[Merged by Bors] - feat: port Probability.ConditionalExpectation

Mathlib.lean

@@ -2424,6 +2424,7 @@ import Mathlib.Order.WithBot
 import Mathlib.Order.Zorn
 import Mathlib.Order.ZornAtoms
 import Mathlib.Probability.CondCount
+import Mathlib.Probability.ConditionalExpectation
 import Mathlib.Probability.ConditionalProbability
 import Mathlib.Probability.Density
 import Mathlib.Probability.Independence.Basic

Mathlib/Probability/ConditionalExpectation.lean

@@ -0,0 +1,82 @@
+/-
+Copyright (c) 2022 Kexing Ying. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kexing Ying
+
+! This file was ported from Lean 3 source module probability.conditional_expectation
+! leanprover-community/mathlib commit 2f8347015b12b0864dfaf366ec4909eb70c78740
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Probability.Notation
+import Mathlib.Probability.Independence.Basic
+import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
+
+/-!
+
+# Probabilistic properties of the conditional expectation
+
+This file contains some properties about the conditional expectation which does not belong in
+the main conditional expectation file.
+
+## Main result
+
+* `MeasureTheory.condexp_indep_eq`: If `m₁, m₂` are independent σ-algebras and `f` is a
+  `m₁`-measurable function, then `𝔼[f | m₂] = 𝔼[f]` almost everywhere.
+
+-/
+
+
+open TopologicalSpace Filter
+
+open scoped NNReal ENNReal MeasureTheory ProbabilityTheory BigOperators
+
+namespace MeasureTheory
+
+open ProbabilityTheory
+
+variable {Ω E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  {m₁ m₂ m : MeasurableSpace Ω} {μ : Measure Ω} {f : Ω → E}
+
+/-- If `m₁, m₂` are independent σ-algebras and `f` is `m₁`-measurable, then `𝔼[f | m₂] = 𝔼[f]`
+almost everywhere. -/
+theorem condexp_indep_eq (hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [SigmaFinite (μ.trim hle₂)]
+    (hf : StronglyMeasurable[m₁] f) (hindp : Indep m₁ m₂ μ) : μ[f|m₂] =ᵐ[μ] fun _ => μ[f] := by
+  by_cases hfint : Integrable f μ
+  swap; · rw [condexp_undef hfint, integral_undef hfint]; rfl
+  refine' (ae_eq_condexp_of_forall_set_integral_eq hle₂ hfint
+    (fun s _ hs => integrableOn_const.2 (Or.inr hs)) (fun s hms hs => _)
+      stronglyMeasurable_const.aeStronglyMeasurable').symm
+  rw [set_integral_const]
+  rw [← memℒp_one_iff_integrable] at hfint
+  refine' Memℒp.induction_stronglyMeasurable hle₁ ENNReal.one_ne_top _ _ _ _ hfint _
+  · exact ⟨f, hf, EventuallyEq.rfl⟩
+  · intro c t hmt _
+    rw [integral_indicator (hle₁ _ hmt), set_integral_const, smul_smul, ← ENNReal.toReal_mul,
+      mul_comm, ← hindp _ _ hmt hms, set_integral_indicator (hle₁ _ hmt), set_integral_const,
+      Set.inter_comm]
+  · intro u v _ huint hvint hu hv hu_eq hv_eq
+    rw [memℒp_one_iff_integrable] at huint hvint
+    rw [integral_add' huint hvint, smul_add, hu_eq, hv_eq,
+      integral_add' huint.integrableOn hvint.integrableOn]
+  · have heq₁ : (fun f : lpMeas E ℝ m₁ 1 μ => ∫ x, (f : Ω → E) x ∂μ) =
+        (fun f : Lp E 1 μ => ∫ x, f x ∂μ) ∘ Submodule.subtypeL _ := by
+      refine' funext fun f => integral_congr_ae _
+      simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]; norm_cast
+    have heq₂ : (fun f : lpMeas E ℝ m₁ 1 μ => ∫ x in s, (f : Ω → E) x ∂μ) =
+        (fun f : Lp E 1 μ => ∫ x in s, f x ∂μ) ∘ Submodule.subtypeL _ := by
+      refine' funext fun f => integral_congr_ae (ae_restrict_of_ae _)
+      simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]
+      exact eventually_of_forall fun _ => (by trivial)
+    refine' isClosed_eq (Continuous.const_smul _ _) _
+    · rw [heq₁]
+      exact continuous_integral.comp (ContinuousLinearMap.continuous _)
+    · rw [heq₂]
+      exact (continuous_set_integral _).comp (ContinuousLinearMap.continuous _)
+  · intro u v huv _ hueq
+    rwa [← integral_congr_ae huv, ←
+      (set_integral_congr_ae (hle₂ _ hms) _ : ∫ x in s, u x ∂μ = ∫ x in s, v x ∂μ)]
+    filter_upwards [huv] with x hx _ using hx
+#align measure_theory.condexp_indep_eq MeasureTheory.condexp_indep_eq
+
+end MeasureTheory