Documentation

Mathlib.MeasureTheory.Integral.SetToL1

Extension of a linear function from indicators to L1 #

Let T : Set α → E →L[ℝ] F be additive for measurable sets with finite measure, in the sense that for s, t two such sets, Disjoint s t → T (s ∪ t) = T s + T t. T is akin to a bilinear map on Set α × E, or a linear map on indicator functions.

This file constructs an extension of T to integrable simple functions, which are finite sums of indicators of measurable sets with finite measure, then to integrable functions, which are limits of integrable simple functions.

The main result is a continuous linear map (α →₁[μ] E) →L[ℝ] F. This extension process is used to define the Bochner integral in the Mathlib.MeasureTheory.Integral.Bochner file and the conditional expectation of an integrable function in Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.

Main Definitions #

Properties #

For most properties of setToFun, we provide two lemmas. One version uses hypotheses valid on all sets, like T = T', and a second version which uses a primed name uses hypotheses on measurable sets with finite measure, like ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s.

The lemmas listed here don't show all hypotheses. Refer to the actual lemmas for details.

Linearity:

Other:

If the space is a NormedLatticeAddCommGroup and T is such that 0 ≤ T s x for 0 ≤ x, we also prove order-related properties:

Implementation notes #

The starting object T : Set α → E →L[ℝ] F matters only through its restriction on measurable sets with finite measure. Its value on other sets is ignored.

def MeasureTheory.FinMeasAdditive {α : Type u_1} {β : Type u_7} [AddMonoid β] {x✝ : MeasurableSpace α} (μ : Measure α) (T : Set αβ) :

A set function is FinMeasAdditive if its value on the union of two disjoint measurable sets with finite measure is the sum of its values on each set.

Equations
theorem MeasureTheory.FinMeasAdditive.zero {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] :
theorem MeasureTheory.FinMeasAdditive.add {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T T' : Set αβ} (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T')
theorem MeasureTheory.FinMeasAdditive.smul {α : Type u_1} {𝕜 : Type u_6} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set αβ} [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
FinMeasAdditive μ fun (s : Set α) => c T s
theorem MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set αβ} {μ' : Measure α} (h : ∀ (s : Set α), MeasurableSet sμ s = μ' s = ) (hT : FinMeasAdditive μ T) :
theorem MeasureTheory.FinMeasAdditive.of_smul_measure {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set αβ} {c : ENNReal} (hc_ne_top : c ) (hT : FinMeasAdditive (c μ) T) :
theorem MeasureTheory.FinMeasAdditive.smul_measure {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set αβ} (c : ENNReal) (hc_ne_zero : c 0) (hT : FinMeasAdditive μ T) :
theorem MeasureTheory.FinMeasAdditive.smul_measure_iff {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set αβ} (c : ENNReal) (hc_ne_zero : c 0) (hc_ne_top : c ) :
theorem MeasureTheory.FinMeasAdditive.map_empty_eq_zero {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_8} [AddCancelMonoid β] {T : Set αβ} (hT : FinMeasAdditive μ T) :
T = 0
theorem MeasureTheory.FinMeasAdditive.map_iUnion_fin_meas_set_eq_sum {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [AddCommMonoid β] (T : Set αβ) (T_empty : T = 0) (h_add : FinMeasAdditive μ T) {ι : Type u_8} (S : ιSet α) ( : Finset ι) (hS_meas : ∀ (i : ι), MeasurableSet (S i)) (hSp : i, μ (S i) ) (h_disj : i, j, i jDisjoint (S i) (S j)) :
T (⋃ i, S i) = i, T (S i)
def MeasureTheory.DominatedFinMeasAdditive {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {x✝ : MeasurableSpace α} (μ : Measure α) (T : Set αβ) (C : ) :

A FinMeasAdditive set function whose norm on every set is less than the measure of the set (up to a multiplicative constant).

Equations
theorem MeasureTheory.DominatedFinMeasAdditive.eq_zero_of_measure_zero {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_8} [NormedAddCommGroup β] {T : Set αβ} {C : } (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hs_zero : μ s = 0) :
T s = 0
theorem MeasureTheory.DominatedFinMeasAdditive.eq_zero {α : Type u_1} {β : Type u_8} [NormedAddCommGroup β] {T : Set αβ} {C : } {x✝ : MeasurableSpace α} (hT : DominatedFinMeasAdditive 0 T C) {s : Set α} (hs : MeasurableSet s) :
T s = 0
theorem MeasureTheory.DominatedFinMeasAdditive.add {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T T' : Set αβ} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') :
DominatedFinMeasAdditive μ (T + T') (C + C')
theorem MeasureTheory.DominatedFinMeasAdditive.smul {α : Type u_1} {𝕜 : Type u_6} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } [NormedField 𝕜] [NormedSpace 𝕜 β] (hT : DominatedFinMeasAdditive μ T C) (c : 𝕜) :
DominatedFinMeasAdditive μ (fun (s : Set α) => c T s) (c * C)
theorem MeasureTheory.DominatedFinMeasAdditive.of_measure_le {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } {μ' : Measure α} (h : μ μ') (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) :
theorem MeasureTheory.DominatedFinMeasAdditive.add_measure_right {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } {x✝ : MeasurableSpace α} (μ ν : Measure α) (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) :
theorem MeasureTheory.DominatedFinMeasAdditive.add_measure_left {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } {x✝ : MeasurableSpace α} (μ ν : Measure α) (hT : DominatedFinMeasAdditive ν T C) (hC : 0 C) :
theorem MeasureTheory.DominatedFinMeasAdditive.of_smul_measure {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } {c : ENNReal} (hc_ne_top : c ) (hT : DominatedFinMeasAdditive (c μ) T C) :
theorem MeasureTheory.DominatedFinMeasAdditive.of_measure_le_smul {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set αβ} {C : } {μ' : Measure α} {c : ENNReal} (hc : c ) (h : μ c μ') (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) :
def MeasureTheory.SimpleFunc.setToSimpleFunc {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] {x✝ : MeasurableSpace α} (T : Set αF →L[] F') (f : SimpleFunc α F) :
F'

Extend Set α → (F →L[ℝ] F') to (α →ₛ F) → F'.

Equations
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_zero' {α : Type u_1} {E : Type u_2} {F' : Type u_4} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F'] [NormedSpace F'] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F'} (h_zero : ∀ (s : Set α), MeasurableSet sμ s < T s = 0) (f : SimpleFunc α E) (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] [DecidablePred fun (x : F) => x 0] {m : MeasurableSpace α} (T : Set αF →L[] F') (f : SimpleFunc α F) :
setToSimpleFunc T f = x{xf.range | x 0}, (T (f ⁻¹' {x})) x
theorem MeasureTheory.SimpleFunc.map_setToSimpleFunc {α : Type u_1} {F : Type u_3} {F' : Type u_4} {G : Type u_5} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] [NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α} (T : Set αF →L[] F') (h_add : FinMeasAdditive μ T) {f : SimpleFunc α G} (hf : Integrable (⇑f) μ) {g : GF} (hg : g 0 = 0) :
setToSimpleFunc T (map g f) = xf.range, (T (f ⁻¹' {x})) (g x)
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_congr' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) {f g : SimpleFunc α E} (hf : Integrable (⇑f) μ) (hg : Integrable (⇑g) μ) (h : Pairwise fun (x y : E) => T (f ⁻¹' {x} g ⁻¹' {y}) = 0) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_congr {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) {f g : SimpleFunc α E} (hf : Integrable (⇑f) μ) (h : f =ᶠ[ae μ] g) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_congr_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' : Set αE →L[] F) (h : ∀ (s : Set α), MeasurableSet sμ s < T s = T' s) (f : SimpleFunc α E) (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_add_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' T'' : Set αE →L[] F) (h_add : ∀ (s : Set α), MeasurableSet sμ s < T'' s = T s + T' s) {f : SimpleFunc α E} (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] {m : MeasurableSpace α} (T : Set αF →L[] F') (c : ) (f : SimpleFunc α F) :
setToSimpleFunc (fun (s : Set α) => c T s) f = c setToSimpleFunc T f
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left' {α : Type u_1} {E : Type u_2} {F' : Type u_4} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F'] [NormedSpace F'] {m : MeasurableSpace α} {μ : Measure α} (T T' : Set αE →L[] F') (c : ) (h_smul : ∀ (s : Set α), MeasurableSet sμ s < T' s = c T s) {f : SimpleFunc α E} (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_add {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) {f g : SimpleFunc α E} (hf : Integrable (⇑f) μ) (hg : Integrable (⇑g) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_neg {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) {f : SimpleFunc α E} (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_sub {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) {f g : SimpleFunc α E} (hf : Integrable (⇑f) μ) (hg : Integrable (⇑g) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) (c : ) {f : SimpleFunc α E} (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_smul {α : Type u_1} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {E : Type u_7} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace E] [NormedSpace 𝕜 F] (T : Set αE →L[] F) (h_add : FinMeasAdditive μ T) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) (c : 𝕜) {f : SimpleFunc α E} (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace F] {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] {m : MeasurableSpace α} (T T' : Set αF →L[] G'') (hTT' : ∀ (s : Set α) (x : F), (T s) x (T' s) x) (f : SimpleFunc α F) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left' {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] (T T' : Set αE →L[] G'') (hTT' : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : E), (T s) x (T' s) x) (f : SimpleFunc α E) (hf : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg {α : Type u_1} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {m : MeasurableSpace α} (T : Set αG' →L[] G'') (hT_nonneg : ∀ (s : Set α) (x : G'), 0 x0 (T s) x) (f : SimpleFunc α G') (hf : 0 f) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg' {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] (T : Set αG' →L[] G'') (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) (f : SimpleFunc α G') (hf : 0 f) (hfi : Integrable (⇑f) μ) :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_mono {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f g : SimpleFunc α G'} (hfi : Integrable (⇑f) μ) (hgi : Integrable (⇑g) μ) (hfg : f g) :
theorem MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] {m : MeasurableSpace α} {μ : Measure α} (T : Set αF →L[] F') {C : } (hT_norm : ∀ (s : Set α), MeasurableSet sT s C * (μ s).toReal) (f : SimpleFunc α F) :
setToSimpleFunc T f C * xf.range, (μ (f ⁻¹' {x})).toReal * x
theorem MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable {α : Type u_1} {E : Type u_2} {F' : Type u_4} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F'] [NormedSpace F'] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F') {C : } (hT_norm : ∀ (s : Set α), MeasurableSet sμ s < T s C * (μ s).toReal) (f : SimpleFunc α E) (hf : Integrable (⇑f) μ) :
setToSimpleFunc T f C * xf.range, (μ (f ⁻¹' {x})).toReal * x
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_indicator {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] (T : Set αF →L[] F') (hT_empty : T = 0) {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (x : F) :
setToSimpleFunc T (piecewise s hs (const α x) (const α 0)) = (T s) x
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_const' {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] [Nonempty α] (T : Set αF →L[] F') (x : F) {m : MeasurableSpace α} :
theorem MeasureTheory.SimpleFunc.setToSimpleFunc_const {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace F] [NormedAddCommGroup F'] [NormedSpace F'] (T : Set αF →L[] F') (hT_empty : T = 0) (x : F) {m : MeasurableSpace α} :
def MeasureTheory.L1.SimpleFunc.setToL1S {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (f : (Lp.simpleFunc E 1 μ)) :
F

Extend Set α → (E →L[ℝ] F') to (α →₁ₛ[μ] E) → F'.

Equations
@[simp]
theorem MeasureTheory.L1.SimpleFunc.setToL1S_zero_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (f : (Lp.simpleFunc E 1 μ)) :
setToL1S 0 f = 0
theorem MeasureTheory.L1.SimpleFunc.setToL1S_zero_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} (h_zero : ∀ (s : Set α), MeasurableSet sμ s < T s = 0) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S T f = 0
theorem MeasureTheory.L1.SimpleFunc.setToL1S_congr {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) {f g : (Lp.simpleFunc E 1 μ)} (h : (Lp.simpleFunc.toSimpleFunc f) =ᶠ[ae μ] (Lp.simpleFunc.toSimpleFunc g)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_congr_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' : Set αE →L[] F) (h : ∀ (s : Set α), MeasurableSet sμ s < T s = T' s) (f : (Lp.simpleFunc E 1 μ)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_congr_measure {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ μ' : Measure α} (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) ( : μ.AbsolutelyContinuous μ') (f : (Lp.simpleFunc E 1 μ)) (f' : (Lp.simpleFunc E 1 μ')) (h : f =ᶠ[ae μ] f') :

setToL1S does not change if we replace the measure μ by μ' with μ ≪ μ'. The statement uses two functions f and f' because they have to belong to different types, but morally these are the same function (we have f =ᵐ[μ] f').

theorem MeasureTheory.L1.SimpleFunc.setToL1S_add_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' : Set αE →L[] F) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S (T + T') f = setToL1S T f + setToL1S T' f
theorem MeasureTheory.L1.SimpleFunc.setToL1S_add_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' T'' : Set αE →L[] F) (h_add : ∀ (s : Set α), MeasurableSet sμ s < T'' s = T s + T' s) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S T'' f = setToL1S T f + setToL1S T' f
theorem MeasureTheory.L1.SimpleFunc.setToL1S_smul_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (c : ) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S (fun (s : Set α) => c T s) f = c setToL1S T f
theorem MeasureTheory.L1.SimpleFunc.setToL1S_smul_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T T' : Set αE →L[] F) (c : ) (h_smul : ∀ (s : Set α), MeasurableSet sμ s < T' s = c T s) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S T' f = c setToL1S T f
theorem MeasureTheory.L1.SimpleFunc.setToL1S_add {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (f g : (Lp.simpleFunc E 1 μ)) :
setToL1S T (f + g) = setToL1S T f + setToL1S T g
theorem MeasureTheory.L1.SimpleFunc.setToL1S_neg {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (f : (Lp.simpleFunc E 1 μ)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_sub {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (f g : (Lp.simpleFunc E 1 μ)) :
setToL1S T (f - g) = setToL1S T f - setToL1S T g
theorem MeasureTheory.L1.SimpleFunc.setToL1S_smul_real {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (c : ) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S T (c f) = c setToL1S T f
theorem MeasureTheory.L1.SimpleFunc.setToL1S_smul {α : Type u_1} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [NormedField 𝕜] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace E] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (T : Set αE →L[] F) (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) (c : 𝕜) (f : (Lp.simpleFunc E 1 μ)) :
setToL1S T (c f) = c setToL1S T f
theorem MeasureTheory.L1.SimpleFunc.norm_setToL1S_le {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} (T : Set αE →L[] F) {C : } (hT_norm : ∀ (s : Set α), MeasurableSet sμ s < T s C * (μ s).toReal) (f : (Lp.simpleFunc E 1 μ)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {s : Set α} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (hs : MeasurableSet s) (hμs : μ s < ) (x : E) :
setToL1S T (Lp.simpleFunc.indicatorConst 1 hs x) = (T s) x
theorem MeasureTheory.L1.SimpleFunc.setToL1S_const {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ] {T : Set αE →L[] F} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (x : E) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_mono_left {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_7} [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T T' : Set αE →L[] G''} (hTT' : ∀ (s : Set α) (x : E), (T s) x (T' s) x) (f : (Lp.simpleFunc E 1 μ)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_mono_left' {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_7} [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T T' : Set αE →L[] G''} (hTT' : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : E), (T s) x (T' s) x) (f : (Lp.simpleFunc E 1 μ)) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_7} {G' : Type u_8} [NormedLatticeAddCommGroup G'] [NormedSpace G'] [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T : Set αG'' →L[] G'} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G''), 0 x0 (T s) x) {f : (Lp.simpleFunc G'' 1 μ)} (hf : 0 f) :
theorem MeasureTheory.L1.SimpleFunc.setToL1S_mono {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_7} {G' : Type u_8} [NormedLatticeAddCommGroup G'] [NormedSpace G'] [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T : Set αG'' →L[] G'} (h_zero : ∀ (s : Set α), MeasurableSet sμ s = 0T s = 0) (h_add : FinMeasAdditive μ T) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G''), 0 x0 (T s) x) {f g : (Lp.simpleFunc G'' 1 μ)} (hfg : f g) :
def MeasureTheory.L1.SimpleFunc.setToL1SCLM' (α : Type u_1) (E : Type u_2) {F : Type u_3} (𝕜 : Type u_6) [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} (μ : Measure α) [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) :
(Lp.simpleFunc E 1 μ) →L[𝕜] F

Extend Set α → E →L[ℝ] F to (α →₁ₛ[μ] E) →L[𝕜] F.

Equations
def MeasureTheory.L1.SimpleFunc.setToL1SCLM (α : Type u_1) (E : Type u_2) {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} (μ : Measure α) {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
(Lp.simpleFunc E 1 μ) →L[] F

Extend Set α → E →L[ℝ] F to (α →₁ₛ[μ] E) →L[ℝ] F.

Equations
@[simp]
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {C : } (hT : DominatedFinMeasAdditive μ 0 C) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f = 0
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ (s : Set α), MeasurableSet sμ s < T s = 0) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f = 0
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f = (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ (s : Set α), MeasurableSet sμ s < T s = T' s) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f = (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_measure {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {C C' : } {μ' : Measure α} (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') ( : μ.AbsolutelyContinuous μ') (f : (Lp.simpleFunc E 1 μ)) (f' : (Lp.simpleFunc E 1 μ')) (h : f =ᶠ[ae μ] f') :
(setToL1SCLM α E μ hT) f = (setToL1SCLM α E μ' hT') f'
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ ) f = (setToL1SCLM α E μ hT) f + (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T T' T'' : Set αE →L[] F} {C C' C'' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ (s : Set α), MeasurableSet sμ s < T'' s = T s + T' s) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT'') f = (setToL1SCLM α E μ hT) f + (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {C : } (c : ) (hT : DominatedFinMeasAdditive μ T C) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ ) f = c (setToL1SCLM α E μ hT) f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T T' : Set αE →L[] F} {C C' : } (c : ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h_smul : ∀ (s : Set α), MeasurableSet sμ s < T' s = c T s) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT') f = c (setToL1SCLM α E μ hT) f
theorem MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) :
setToL1SCLM α E μ hT C
theorem MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
setToL1SCLM α E μ hT C 0
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_const {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [IsFiniteMeasure μ] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (x : E) :
(setToL1SCLM α E μ hT) (Lp.simpleFunc.indicatorConst 1 x) = (T Set.univ) x
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α) (x : E), (T s) x (T' s) x) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left' {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : E), (T s) x (T' s) x) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1SCLM α E μ hT) f (setToL1SCLM α E μ hT') f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f : (Lp.simpleFunc G' 1 μ)} (hf : 0 f) :
0 (setToL1SCLM α G' μ hT) f
theorem MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f g : (Lp.simpleFunc G' 1 μ)} (hfg : f g) :
(setToL1SCLM α G' μ hT) f (setToL1SCLM α G' μ hT) g
def MeasureTheory.L1.setToL1' {α : Type u_1} {E : Type u_2} {F : Type u_3} (𝕜 : Type u_6) [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) :
(Lp E 1 μ) →L[𝕜] F

Extend Set α → (E →L[ℝ] F) to (α →₁[μ] E) →L[𝕜] F.

Equations
def MeasureTheory.L1.setToL1 {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
(Lp E 1 μ) →L[] F

Extend Set α → E →L[ℝ] F to (α →₁[μ] E) →L[ℝ] F.

Equations
theorem MeasureTheory.L1.setToL1_eq_setToL1SCLM {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp.simpleFunc E 1 μ)) :
(setToL1 hT) f = (SimpleFunc.setToL1SCLM α E μ hT) f
theorem MeasureTheory.L1.setToL1_eq_setToL1' {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) (f : (Lp E 1 μ)) :
(setToL1 hT) f = (setToL1' 𝕜 hT h_smul) f
@[simp]
theorem MeasureTheory.L1.setToL1_zero_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {C : } (hT : DominatedFinMeasAdditive μ 0 C) (f : (Lp E 1 μ)) :
(setToL1 hT) f = 0
theorem MeasureTheory.L1.setToL1_zero_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ (s : Set α), MeasurableSet sμ s < T s = 0) (f : (Lp E 1 μ)) :
(setToL1 hT) f = 0
theorem MeasureTheory.L1.setToL1_congr_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] (T T' : Set αE →L[] F) {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : (Lp E 1 μ)) :
(setToL1 hT) f = (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_congr_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] (T T' : Set αE →L[] F) {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ (s : Set α), MeasurableSet sμ s < T s = T' s) (f : (Lp E 1 μ)) :
(setToL1 hT) f = (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_add_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : (Lp E 1 μ)) :
(setToL1 ) f = (setToL1 hT) f + (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_add_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' T'' : Set αE →L[] F} {C C' C'' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ (s : Set α), MeasurableSet sμ s < T'' s = T s + T' s) (f : (Lp E 1 μ)) :
(setToL1 hT'') f = (setToL1 hT) f + (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_smul_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (c : ) (f : (Lp E 1 μ)) :
(setToL1 ) f = c (setToL1 hT) f
theorem MeasureTheory.L1.setToL1_smul_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (c : ) (h_smul : ∀ (s : Set α), MeasurableSet sμ s < T' s = c T s) (f : (Lp E 1 μ)) :
(setToL1 hT') f = c (setToL1 hT) f
theorem MeasureTheory.L1.setToL1_smul {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) (c : 𝕜) (f : (Lp E 1 μ)) :
(setToL1 hT) (c f) = c (setToL1 hT) f
theorem MeasureTheory.L1.setToL1_simpleFunc_indicatorConst {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ) (x : E) :
(setToL1 hT) (Lp.simpleFunc.indicatorConst 1 hs x) = (T s) x
theorem MeasureTheory.L1.setToL1_indicatorConstLp {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ) (x : E) :
(setToL1 hT) (indicatorConstLp 1 hs hμs x) = (T s) x
theorem MeasureTheory.L1.setToL1_const {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) :
(setToL1 hT) (indicatorConstLp 1 x) = (T Set.univ) x
theorem MeasureTheory.L1.setToL1_mono_left' {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : E), (T s) x (T' s) x) (f : (Lp E 1 μ)) :
(setToL1 hT) f (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_mono_left {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α) (x : E), (T s) x (T' s) x) (f : (Lp E 1 μ)) :
(setToL1 hT) f (setToL1 hT') f
theorem MeasureTheory.L1.setToL1_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f : (Lp G' 1 μ)} (hf : 0 f) :
0 (setToL1 hT) f
theorem MeasureTheory.L1.setToL1_mono {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f g : (Lp G' 1 μ)} (hfg : f g) :
(setToL1 hT) f (setToL1 hT) g
theorem MeasureTheory.L1.norm_setToL1_le_mul_norm {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) (f : (Lp E 1 μ)) :
theorem MeasureTheory.L1.norm_setToL1_le_mul_norm' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp E 1 μ)) :
(setToL1 hT) f (C 0) * f
theorem MeasureTheory.L1.norm_setToL1_le {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (hC : 0 C) :
theorem MeasureTheory.L1.norm_setToL1_le' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
theorem MeasureTheory.L1.tendsto_setToL1 {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp E 1 μ)) {ι : Type u_7} (fs : ι(Lp E 1 μ)) {l : Filter ι} (hfs : Filter.Tendsto fs l (nhds f)) :
Filter.Tendsto (fun (i : ι) => (setToL1 hT) (fs i)) l (nhds ((setToL1 hT) f))

If fs i → f in L1, then setToL1 hT (fs i) → setToL1 hT f.

def MeasureTheory.setToFun {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} (μ : Measure α) [CompleteSpace F] (T : Set αE →L[] F) {C : } (hT : DominatedFinMeasAdditive μ T C) (f : αE) :
F

Extend T : Set α → E →L[ℝ] F to (α → E) → F (for integrable functions α → E). We set it to 0 if the function is not integrable.

Equations
theorem MeasureTheory.setToFun_eq {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
setToFun μ T hT f = (L1.setToL1 hT) (Integrable.toL1 f hf)
theorem MeasureTheory.L1.setToFun_eq_setToL1 {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp E 1 μ)) :
setToFun μ T hT f = (setToL1 hT) f
theorem MeasureTheory.setToFun_undef {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : ¬Integrable f μ) :
setToFun μ T hT f = 0
theorem MeasureTheory.setToFun_non_aEStronglyMeasurable {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : ¬AEStronglyMeasurable f μ) :
setToFun μ T hT f = 0
theorem MeasureTheory.setToFun_congr_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : αE) :
setToFun μ T hT f = setToFun μ T' hT' f
theorem MeasureTheory.setToFun_congr_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ (s : Set α), MeasurableSet sμ s < T s = T' s) (f : αE) :
setToFun μ T hT f = setToFun μ T' hT' f
theorem MeasureTheory.setToFun_add_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (f : αE) :
setToFun μ (T + T') f = setToFun μ T hT f + setToFun μ T' hT' f
theorem MeasureTheory.setToFun_add_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' T'' : Set αE →L[] F} {C C' C'' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ (s : Set α), MeasurableSet sμ s < T'' s = T s + T' s) (f : αE) :
setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f
theorem MeasureTheory.setToFun_smul_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (c : ) (f : αE) :
setToFun μ (fun (s : Set α) => c T s) f = c setToFun μ T hT f
theorem MeasureTheory.setToFun_smul_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T T' : Set αE →L[] F} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (c : ) (h_smul : ∀ (s : Set α), MeasurableSet sμ s < T' s = c T s) (f : αE) :
setToFun μ T' hT' f = c setToFun μ T hT f
@[simp]
theorem MeasureTheory.setToFun_zero {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
setToFun μ T hT 0 = 0
@[simp]
theorem MeasureTheory.setToFun_zero_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {C : } {f : αE} {hT : DominatedFinMeasAdditive μ 0 C} :
setToFun μ 0 hT f = 0
theorem MeasureTheory.setToFun_zero_left' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (h_zero : ∀ (s : Set α), MeasurableSet sμ s < T s = 0) :
setToFun μ T hT f = 0
theorem MeasureTheory.setToFun_add {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f g : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hg : Integrable g μ) :
setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT g
theorem MeasureTheory.setToFun_finset_sum' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {ι : Type u_7} (s : Finset ι) {f : ιαE} (hf : is, Integrable (f i) μ) :
setToFun μ T hT (∑ is, f i) = is, setToFun μ T hT (f i)
theorem MeasureTheory.setToFun_finset_sum {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {ι : Type u_7} (s : Finset ι) {f : ιαE} (hf : is, Integrable (f i) μ) :
(setToFun μ T hT fun (a : α) => is, f i a) = is, setToFun μ T hT (f i)
theorem MeasureTheory.setToFun_neg {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : αE) :
setToFun μ T hT (-f) = -setToFun μ T hT f
theorem MeasureTheory.setToFun_sub {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f g : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hg : Integrable g μ) :
setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g
theorem MeasureTheory.setToFun_smul {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c x) = c (T s) x) (c : 𝕜) (f : αE) :
setToFun μ T hT (c f) = c setToFun μ T hT f
theorem MeasureTheory.setToFun_congr_ae {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f g : αE} (hT : DominatedFinMeasAdditive μ T C) (h : f =ᶠ[ae μ] g) :
setToFun μ T hT f = setToFun μ T hT g
theorem MeasureTheory.setToFun_measure_zero {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (h : μ = 0) :
setToFun μ T hT f = 0
theorem MeasureTheory.setToFun_measure_zero' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (h : ∀ (s : Set α), MeasurableSet sμ s < μ s = 0) :
setToFun μ T hT f = 0
theorem MeasureTheory.setToFun_toL1 {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
setToFun μ T hT (Integrable.toL1 f hf) = setToFun μ T hT f
theorem MeasureTheory.setToFun_indicator_const {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ) (x : E) :
setToFun μ T hT (s.indicator fun (x_1 : α) => x) = (T s) x
theorem MeasureTheory.setToFun_const {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) :
(setToFun μ T hT fun (x_1 : α) => x) = (T Set.univ) x
theorem MeasureTheory.setToFun_mono_left' {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : E), (T s) x (T' s) x) (f : αE) :
setToFun μ T hT f setToFun μ T' hT' f
theorem MeasureTheory.setToFun_mono_left {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] {m : MeasurableSpace α} {μ : Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] {T T' : Set αE →L[] G''} {C C' : } (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α) (x : E), (T s) x (T' s) x) (f : (Lp E 1 μ)) :
setToFun μ T hT f setToFun μ T' hT' f
theorem MeasureTheory.setToFun_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f : αG'} (hf : 0 ≤ᶠ[ae μ] f) :
0 setToFun μ T hT f
theorem MeasureTheory.setToFun_mono {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace G'] {T : Set αG' →L[] G''} {C : } (hT : DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet sμ s < ∀ (x : G'), 0 x0 (T s) x) {f g : αG'} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᶠ[ae μ] g) :
setToFun μ T hT f setToFun μ T hT g
theorem MeasureTheory.continuous_setToFun {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) :
Continuous fun (f : (Lp E 1 μ)) => setToFun μ T hT f
theorem MeasureTheory.tendsto_setToFun_of_L1 {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {ι : Type u_7} (f : αE) (hfi : Integrable f μ) {fs : ιαE} {l : Filter ι} (hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ) (hfs : Filter.Tendsto (fun (i : ι) => ∫⁻ (x : α), fs i x - f x‖ₑ μ) l (nhds 0)) :
Filter.Tendsto (fun (i : ι) => setToFun μ T hT (fs i)) l (nhds (setToFun μ T hT f))

If F i → f in L1, then setToFun μ T hT (F i) → setToFun μ T hT f.

theorem MeasureTheory.tendsto_setToFun_approxOn_of_measurable {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : αE} {s : Set E} [TopologicalSpace.SeparableSpace s] (hfi : Integrable f μ) (hfm : Measurable f) (hs : ∀ᵐ (x : α) μ, f x closure s) {y₀ : E} (h₀ : y₀ s) (h₀i : Integrable (fun (x : α) => y₀) μ) :
Filter.Tendsto (fun (n : ) => setToFun μ T hT (SimpleFunc.approxOn f hfm s y₀ h₀ n)) Filter.atTop (nhds (setToFun μ T hT f))
theorem MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : αE} (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [TopologicalSpace.SeparableSpace s] (hs : Set.range f {0} s) :
Filter.Tendsto (fun (n : ) => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 n)) Filter.atTop (nhds (setToFun μ T hT f))
theorem MeasureTheory.continuous_L1_toL1 {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m : MeasurableSpace α} {μ μ' : Measure α} (c' : ENNReal) (hc' : c' ) (hμ'_le : μ' c' μ) :
Continuous fun (f : (Lp G 1 μ)) => Integrable.toL1 f

Auxiliary lemma for setToFun_congr_measure: the function sending f : α →₁[μ] G to f : α →₁[μ'] G is continuous when μ' ≤ c' • μ for c' ≠ ∞.

theorem MeasureTheory.setToFun_congr_measure_of_integrable {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C C' : } {μ' : Measure α} (c' : ENNReal) (hc' : c' ) (hμ'_le : μ' c' μ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') (f : αE) (hfμ : Integrable f μ) :
setToFun μ T hT f = setToFun μ' T hT' f
theorem MeasureTheory.setToFun_congr_measure {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C C' : } {μ' : Measure α} (c c' : ENNReal) (hc : c ) (hc' : c' ) (hμ_le : μ c μ') (hμ'_le : μ' c' μ) (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ' T C') (f : αE) :
setToFun μ T hT f = setToFun μ' T hT' f
theorem MeasureTheory.setToFun_congr_measure_of_add_right {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C C' : } {μ' : Measure α} (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ T C) (f : αE) (hf : Integrable f (μ + μ')) :
setToFun (μ + μ') T hT_add f = setToFun μ T hT f
theorem MeasureTheory.setToFun_congr_measure_of_add_left {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C C' : } {μ' : Measure α} (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ' T C) (f : αE) (hf : Integrable f (μ + μ')) :
setToFun (μ + μ') T hT_add f = setToFun μ' T hT f
theorem MeasureTheory.setToFun_top_smul_measure {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive ( μ) T C) (f : αE) :
setToFun ( μ) T hT f = 0
theorem MeasureTheory.setToFun_congr_smul_measure {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C C' : } (c : ENNReal) (hc_ne_top : c ) (hT : DominatedFinMeasAdditive μ T C) (hT_smul : DominatedFinMeasAdditive (c μ) T C') (f : αE) :
setToFun μ T hT f = setToFun (c μ) T hT_smul f
theorem MeasureTheory.norm_setToFun_le_mul_norm {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp E 1 μ)) (hC : 0 C) :
setToFun μ T hT f C * f
theorem MeasureTheory.norm_setToFun_le_mul_norm' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) (f : (Lp E 1 μ)) :
setToFun μ T hT f (C 0) * f
theorem MeasureTheory.norm_setToFun_le {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hC : 0 C) :
theorem MeasureTheory.norm_setToFun_le' {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {f : αE} (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
theorem MeasureTheory.tendsto_setToFun_of_dominated_convergence {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {fs : αE} {f : αE} (bound : α) (fs_measurable : ∀ (n : ), AEStronglyMeasurable (fs n) μ) (bound_integrable : Integrable bound μ) (h_bound : ∀ (n : ), ∀ᵐ (a : α) μ, fs n a bound a) (h_lim : ∀ᵐ (a : α) μ, Filter.Tendsto (fun (n : ) => fs n a) Filter.atTop (nhds (f a))) :
Filter.Tendsto (fun (n : ) => setToFun μ T hT (fs n)) Filter.atTop (nhds (setToFun μ T hT f))

Lebesgue dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies the convergence of their image by setToFun. We could weaken the condition bound_integrable to require HasFiniteIntegral bound μ instead (i.e. not requiring that bound is measurable), but in all applications proving integrability is easier.

theorem MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } (hT : DominatedFinMeasAdditive μ T C) {ι : Type u_7} {l : Filter ι} [l.IsCountablyGenerated] {fs : ιαE} {f : αE} (bound : α) (hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) μ, fs n a bound a) (bound_integrable : Integrable bound μ) (h_lim : ∀ᵐ (a : α) μ, Filter.Tendsto (fun (n : ι) => fs n a) l (nhds (f a))) :
Filter.Tendsto (fun (n : ι) => setToFun μ T hT (fs n)) l (nhds (setToFun μ T hT f))

Lebesgue dominated convergence theorem for filters with a countable basis

theorem MeasureTheory.continuousWithinAt_setToFun_of_dominated {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : DominatedFinMeasAdditive μ T C) {fs : XαE} {x₀ : X} {bound : α} {s : Set X} (hfs_meas : ∀ᶠ (x : X) in nhdsWithin x₀ s, AEStronglyMeasurable (fs x) μ) (h_bound : ∀ᶠ (x : X) in nhdsWithin x₀ s, ∀ᵐ (a : α) μ, fs x a bound a) (bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ (a : α) μ, ContinuousWithinAt (fun (x : X) => fs x a) s x₀) :
ContinuousWithinAt (fun (x : X) => setToFun μ T hT (fs x)) s x₀
theorem MeasureTheory.continuousAt_setToFun_of_dominated {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : DominatedFinMeasAdditive μ T C) {fs : XαE} {x₀ : X} {bound : α} (hfs_meas : ∀ᶠ (x : X) in nhds x₀, AEStronglyMeasurable (fs x) μ) (h_bound : ∀ᶠ (x : X) in nhds x₀, ∀ᵐ (a : α) μ, fs x a bound a) (bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ (a : α) μ, ContinuousAt (fun (x : X) => fs x a) x₀) :
ContinuousAt (fun (x : X) => setToFun μ T hT (fs x)) x₀
theorem MeasureTheory.continuousOn_setToFun_of_dominated {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : DominatedFinMeasAdditive μ T C) {fs : XαE} {bound : α} {s : Set X} (hfs_meas : xs, AEStronglyMeasurable (fs x) μ) (h_bound : xs, ∀ᵐ (a : α) μ, fs x a bound a) (bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ (a : α) μ, ContinuousOn (fun (x : X) => fs x a) s) :
ContinuousOn (fun (x : X) => setToFun μ T hT (fs x)) s
theorem MeasureTheory.continuous_setToFun_of_dominated {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup F] [NormedSpace F] {m : MeasurableSpace α} {μ : Measure α} [CompleteSpace F] {T : Set αE →L[] F} {C : } {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : DominatedFinMeasAdditive μ T C) {fs : XαE} {bound : α} (hfs_meas : ∀ (x : X), AEStronglyMeasurable (fs x) μ) (h_bound : ∀ (x : X), ∀ᵐ (a : α) μ, fs x a bound a) (bound_integrable : Integrable bound μ) (h_cont : ∀ᵐ (a : α) μ, Continuous fun (x : X) => fs x a) :
Continuous fun (x : X) => setToFun μ T hT (fs x)