Identically distributed random variables

Two random variables defined on two (possibly different) probability spaces but taking value in the same space are identically distributed if their distributions (i.e., the image probability measures on the target space) coincide. We define this concept and establish its basic properties in this file.

Main definitions and results

• IdentDistrib f g μ ν registers that the image of μ under f coincides with the image of ν under g (and that f and g are almost everywhere measurable, as otherwise the image measures don't make sense). The measures can be kept implicit as in IdentDistrib f g if the spaces are registered as measure spaces.
• IdentDistrib.comp: being identically distributed is stable under composition with measurable maps.

There are two main kinds of lemmas, under the assumption that f and g are identically distributed: lemmas saying that two quantities computed for f and g are the same, and lemmas saying that if f has some property then g also has it. The first kind is registered as IdentDistrib.foo_fst, the second one as IdentDistrib.foo_snd (in the latter case, to deduce a property of f from one of g, use h.symm.foo_snd where h : IdentDistrib f g μ ν). For instance:

• IdentDistrib.measure_mem_eq: if f and g are identically distributed, then the probabilities that they belong to a given measurable set are the same.

• IdentDistrib.integral_eq: if f and g are identically distributed, then their integrals are the same.

• IdentDistrib.variance_eq: if f and g are identically distributed, then their variances are the same.

• IdentDistrib.aestronglyMeasurable_snd: if f and g are identically distributed and f is almost everywhere strongly measurable, then so is g.

• IdentDistrib.memLp_snd: if f and g are identically distributed and f belongs to ℒp, then so does g.

We also register several dot notation shortcuts for convenience. For instance, if h : IdentDistrib f g μ ν, then h.sq states that f^2 and g^2 are identically distributed, and h.norm states that ‖f‖ and ‖g‖ are identically distributed, and so on.

structure ProbabilityTheory.IdentDistrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (f : α → γ) (g : β → γ) (μ : MeasureTheory.Measure α := by volume_tac) (ν : MeasureTheory.Measure β := by volume_tac) : Prop

Two functions defined on two (possibly different) measure spaces are identically distributed if their image measures coincide. This only makes sense when the functions are ae measurable (as otherwise the image measures are not defined), so we require this as well in the definition.

• aemeasurable_fst : AEMeasurable f μ
• aemeasurable_snd : AEMeasurable g ν
• map_eq : MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g ν

theorem ProbabilityTheory.IdentDistrib.refl {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → γ} (hf : AEMeasurable f μ) : IdentDistrib f f μ μ

theorem ProbabilityTheory.IdentDistrib.symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ

theorem ProbabilityTheory.IdentDistrib.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {ρ : MeasureTheory.Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν) (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ

theorem ProbabilityTheory.IdentDistrib.comp_of_aemeasurable {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : AEMeasurable u (MeasureTheory.Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν

theorem ProbabilityTheory.IdentDistrib.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) : IdentDistrib (u ∘ f) (u ∘ g) μ ν

theorem ProbabilityTheory.IdentDistrib.of_ae_eq {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) : IdentDistrib f g μ μ

theorem MeasureTheory.AEMeasurable.identDistrib_mk {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : Measure α} {f : α → γ} (hf : AEMeasurable f μ) : ProbabilityTheory.IdentDistrib f (AEMeasurable.mk f hf) μ μ

theorem MeasureTheory.AEStronglyMeasurable.identDistrib_mk {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {μ : Measure α} {f : α → γ} [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] (hf : AEStronglyMeasurable f μ) : ProbabilityTheory.IdentDistrib f (AEStronglyMeasurable.mk f hf) μ μ

theorem ProbabilityTheory.IdentDistrib.measure_mem_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) : μ (f ⁻¹' s) = ν (g ⁻¹' s)

theorem ProbabilityTheory.IdentDistrib.measure_preimage_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) : μ (f ⁻¹' s) = ν (g ⁻¹' s)

Alias of ProbabilityTheory.IdentDistrib.measure_mem_eq.

theorem ProbabilityTheory.IdentDistrib.ae_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x : γ | p x}) (hp : ∀ᵐ (x : α) ∂μ, p (f x)) : ∀ᵐ (x : β) ∂ν, p (g x)

theorem ProbabilityTheory.IdentDistrib.ae_mem_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t) (ht : ∀ᵐ (x : α) ∂μ, f x ∈ t) : ∀ᵐ (x : β) ∂ν, g x ∈ t

theorem ProbabilityTheory.IdentDistrib.aestronglyMeasurable_fst {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) : MeasureTheory.AEStronglyMeasurable f μ

In a second countable topology, the first function in an identically distributed pair is a.e. strongly measurable. So is the second function, but use h.symm.aestronglyMeasurable_fst as h.aestronglyMeasurable_snd has a different meaning.

theorem ProbabilityTheory.IdentDistrib.aestronglyMeasurable_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) (hf : MeasureTheory.AEStronglyMeasurable f μ) : MeasureTheory.AEStronglyMeasurable g ν

If f and g are identically distributed and f is a.e. strongly measurable, so is g.

theorem ProbabilityTheory.IdentDistrib.aestronglyMeasurable_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) : MeasureTheory.AEStronglyMeasurable f μ ↔ MeasureTheory.AEStronglyMeasurable g ν

theorem ProbabilityTheory.IdentDistrib.essSup_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ] [OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν

theorem ProbabilityTheory.IdentDistrib.lintegral_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ENNReal} {g : β → ENNReal} (h : IdentDistrib f g μ ν) : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : β), g x ∂ν

theorem ProbabilityTheory.IdentDistrib.integral_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) : ∫ (x : α), f x ∂μ = ∫ (x : β), g x ∂ν

theorem ProbabilityTheory.IdentDistrib.eLpNorm_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν) (p : ENNReal) : MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm g p ν

theorem ProbabilityTheory.IdentDistrib.memLp_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] {p : ENNReal} (h : IdentDistrib f g μ ν) (hf : MeasureTheory.MemLp f p μ) : MeasureTheory.MemLp g p ν

theorem ProbabilityTheory.IdentDistrib.memLp_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] {p : ENNReal} (h : IdentDistrib f g μ ν) : MeasureTheory.MemLp f p μ ↔ MeasureTheory.MemLp g p ν

theorem ProbabilityTheory.IdentDistrib.integrable_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) (hf : MeasureTheory.Integrable f μ) : MeasureTheory.Integrable g ν

theorem ProbabilityTheory.IdentDistrib.integrable_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) : MeasureTheory.Integrable f μ ↔ MeasureTheory.Integrable g ν

theorem ProbabilityTheory.IdentDistrib.norm {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν) : IdentDistrib (fun (x : α) => ‖f x‖) (fun (x : β) => ‖g x‖) μ ν

theorem ProbabilityTheory.IdentDistrib.nnnorm {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν) : IdentDistrib (fun (x : α) => ‖f x‖₊) (fun (x : β) => ‖g x‖₊) μ ν

theorem ProbabilityTheory.IdentDistrib.pow {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) {n : ℕ} : IdentDistrib (fun (x : α) => f x ^ n) (fun (x : β) => g x ^ n) μ ν

theorem ProbabilityTheory.IdentDistrib.sq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) : IdentDistrib (fun (x : α) => f x ^ 2) (fun (x : β) => g x ^ 2) μ ν

theorem ProbabilityTheory.IdentDistrib.coe_nnreal_ennreal {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → NNReal} {g : β → NNReal} (h : IdentDistrib f g μ ν) : IdentDistrib (fun (x : α) => ↑(f x)) (fun (x : β) => ↑(g x)) μ ν

theorem ProbabilityTheory.IdentDistrib.mul_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => f x * c) (fun (x : β) => g x * c) μ ν

theorem ProbabilityTheory.IdentDistrib.add_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Add γ] [MeasurableAdd γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => f x + c) (fun (x : β) => g x + c) μ ν

theorem ProbabilityTheory.IdentDistrib.const_mul {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => c * f x) (fun (x : β) => c * g x) μ ν

theorem ProbabilityTheory.IdentDistrib.const_add {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Add γ] [MeasurableAdd γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => c + f x) (fun (x : β) => c + g x) μ ν

theorem ProbabilityTheory.IdentDistrib.div_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => f x / c) (fun (x : β) => g x / c) μ ν

theorem ProbabilityTheory.IdentDistrib.sub_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Sub γ] [MeasurableSub γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => f x - c) (fun (x : β) => g x - c) μ ν

theorem ProbabilityTheory.IdentDistrib.const_div {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => c / f x) (fun (x : β) => c / g x) μ ν

theorem ProbabilityTheory.IdentDistrib.const_sub {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Sub γ] [MeasurableSub γ] (h : IdentDistrib f g μ ν) (c : γ) : IdentDistrib (fun (x : α) => c - f x) (fun (x : β) => c - g x) μ ν

theorem ProbabilityTheory.IdentDistrib.inv {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Inv γ] [MeasurableInv γ] (h : IdentDistrib f g μ ν) : IdentDistrib f⁻¹ g⁻¹ μ ν

theorem ProbabilityTheory.IdentDistrib.neg {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} [Neg γ] [MeasurableNeg γ] (h : IdentDistrib f g μ ν) : IdentDistrib (-f) (-g) μ ν

theorem ProbabilityTheory.IdentDistrib.evariance_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) : evariance f μ = evariance g ν

theorem ProbabilityTheory.IdentDistrib.variance_eq {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) : variance f μ = variance g ν

theorem ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib_aux {α : Type u_1} [MeasurableSpace α] {E : Type u_5} [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_6} {f : ι → α → E} {j : ι} {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hℒp : MeasureTheory.MemLp (f j) p μ) (hfmeas : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ) : MeasureTheory.UniformIntegrable f p μ

This lemma is superseded by MemLp.uniformIntegrable_of_identDistrib which only requires AEStronglyMeasurable.

theorem ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib {α : Type u_1} [MeasurableSpace α] {E : Type u_5} [MeasurableSpace E] [NormedAddCommGroup E] [BorelSpace E] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {ι : Type u_6} {f : ι → α → E} {j : ι} {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (hℒp : MeasureTheory.MemLp (f j) p μ) (hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ) : MeasureTheory.UniformIntegrable f p μ

A sequence of identically distributed Lᵖ functions is p-uniformly integrable.

theorem ProbabilityTheory.indepFun_of_identDistrib_pair {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure γ} {μ' : MeasureTheory.Measure δ} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] {X : γ → α} {X' : δ → α} {Y : γ → β} {Y' : δ → β} (h_indep : IndepFun X Y μ) (h_ident : IdentDistrib (fun (ω : γ) => (X ω, Y ω)) (fun (ω : δ) => (X' ω, Y' ω)) μ μ') : IndepFun X' Y' μ'

If X and Y are independent and (X, Y) and (X', Y') are identically distributed, then X' and Y' are independent.