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Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Conditional expectation of indicator functions #

This file proves some results about the conditional expectation of an indicator function and as a corollary, also proves several results about the behaviour of the conditional expectation on a restricted measure.

Main result #

theorem MeasureTheory.condexp_ae_eq_restrict_zero {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : αE} {s : Set α} (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict s] 0) :
MeasureTheory.condexp m μ f =ᵐ[μ.restrict s] 0
theorem MeasureTheory.condexp_indicator_aux {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : αE} {s : Set α} (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict s] 0) :
MeasureTheory.condexp m μ (s.indicator f) =ᵐ[μ] s.indicator (MeasureTheory.condexp m μ f)

Auxiliary lemma for condexp_indicator.

theorem MeasureTheory.condexp_indicator {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : αE} {s : Set α} (hf_int : MeasureTheory.Integrable f μ) (hs : MeasurableSet s) :
MeasureTheory.condexp m μ (s.indicator f) =ᵐ[μ] s.indicator (MeasureTheory.condexp m μ f)

The conditional expectation of the indicator of a function over an m-measurable set with respect to the σ-algebra m is a.e. equal to the indicator of the conditional expectation.

theorem MeasureTheory.condexp_restrict_ae_eq_restrict {α : Type u_1} {E : Type u_3} {m : MeasurableSpace α} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {f : αE} {s : Set α} (hm : m m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs_m : MeasurableSet s) (hf_int : MeasureTheory.Integrable f μ) :
MeasureTheory.condexp m (μ.restrict s) f =ᵐ[μ.restrict s] MeasureTheory.condexp m μ f
theorem MeasureTheory.condexp_ae_eq_restrict_of_measurableSpace_eq_on {α : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {f : αE} {s : Set α} {m : MeasurableSpace α} {m₂ : MeasurableSpace α} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m m0) (hm₂ : m₂ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] [MeasureTheory.SigmaFinite (μ.trim hm₂)] (hs_m : MeasurableSet s) (hs : ∀ (t : Set α), MeasurableSet (s t) MeasurableSet (s t)) :
MeasureTheory.condexp m μ f =ᵐ[μ.restrict s] MeasureTheory.condexp m₂ μ f

If the restriction to an m-measurable set s of a σ-algebra m is equal to the restriction to s of another σ-algebra m₂ (hypothesis hs), then μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂].