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[Merged by Bors] - feat: when the indicator is independent from a random variable #39749

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[Merged by Bors] - feat: when the indicator is independent from a random variable #39749

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6 changes: 6 additions & 0 deletions Mathlib/Probability/Independence/Basic.lean
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Expand Up @@ -1011,6 +1011,12 @@ theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β
iIndepFun (fun n => (s n).indicator fun _ω => (1 : β)) μ :=
Kernel.iIndepSet.iIndepFun_indicator hs

lemma Indep.indicator_indepFun {m : MeasurableSpace Ω} {M 𝓧 : Type*}
[Zero M] [MeasurableSpace M] (c : M) [NeZero c] {m𝓧 : MeasurableSpace 𝓧} {A : Set Ω}
{X : Ω → 𝓧} (hA : MeasurableSet[m] A) (h : Indep m (m𝓧.comap X) μ) :
(A.indicator (fun _ ↦ c)) ⟂ᵢ[μ] X :=
Kernel.Indep.indicator_const_indepFun c hA h

end IndepFun

variable {ι Ω α β : Type*} {mΩ : MeasurableSpace Ω} {mα : MeasurableSpace α}
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6 changes: 6 additions & 0 deletions Mathlib/Probability/Independence/Kernel/IndepFun.lean
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Expand Up @@ -694,6 +694,12 @@ theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β
· exact @MeasurableSet.empty _ (generateFrom {s i})
· exact @MeasurableSet.empty _ (generateFrom {s i})

lemma Indep.indicator_const_indepFun {m : MeasurableSpace Ω} {M 𝓧 : Type*}
[Zero M] [MeasurableSpace M] (c : M) [NeZero c] {m𝓧 : MeasurableSpace 𝓧} {A : Set Ω}
{X : Ω → 𝓧} (hA : MeasurableSet[m] A) (h : Indep m (m𝓧.comap X) κ μ) :
IndepFun (A.indicator (fun _ ↦ c)) X κ μ :=
indep_of_indep_of_le_left h (measurable_const.indicator hA).comap_le

variable {mβ : MeasurableSpace β} {X : ι → Ω → α} {Y : ι → Ω → β}
{f : _ → Set Ω} {t : ι → Set β} {s : Finset ι}

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