Conditional expectation of indicator functions #
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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 #
measure_theory.condexp_indicator: Ifsis am-measurable set, then the conditional expectation of the indicator function ofsis almost everywhere equal to the indicator ofsof the conditional expectation. Namely,𝔼[s.indicator f | m] = s.indicator 𝔼[f | m]a.e.
theorem
measure_theory.condexp_ae_eq_restrict_zero
{α : Type u_1}
{E : Type u_3}
{m m0 : measurable_space α}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
{f : α → E}
{s : set α}
(hs : measurable_set s)
(hf : f =ᵐ[μ.restrict s] 0) :
measure_theory.condexp m μ f =ᵐ[μ.restrict s] 0
theorem
measure_theory.condexp_indicator_aux
{α : Type u_1}
{E : Type u_3}
{m m0 : measurable_space α}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
{f : α → E}
{s : set α}
(hs : measurable_set s)
(hf : f =ᵐ[μ.restrict sᶜ] 0) :
measure_theory.condexp m μ (s.indicator f) =ᵐ[μ] s.indicator (measure_theory.condexp m μ f)
Auxiliary lemma for condexp_indicator.
theorem
measure_theory.condexp_indicator
{α : Type u_1}
{E : Type u_3}
{m m0 : measurable_space α}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
{f : α → E}
{s : set α}
(hf_int : measure_theory.integrable f μ)
(hs : measurable_set s) :
measure_theory.condexp m μ (s.indicator f) =ᵐ[μ] s.indicator (measure_theory.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
measure_theory.condexp_restrict_ae_eq_restrict
{α : Type u_1}
{E : Type u_3}
{m m0 : measurable_space α}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{μ : measure_theory.measure α}
{f : α → E}
{s : set α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(hs_m : measurable_set s)
(hf_int : measure_theory.integrable f μ) :
measure_theory.condexp m (μ.restrict s) f =ᵐ[μ.restrict s] measure_theory.condexp m μ f
theorem
measure_theory.condexp_ae_eq_restrict_of_measurable_space_eq_on
{α : Type u_1}
{E : Type u_3}
[normed_add_comm_group E]
[normed_space ℝ E]
[complete_space E]
{f : α → E}
{s : set α}
{m m₂ m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
(hm₂ : m₂ ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
[measure_theory.sigma_finite (μ.trim hm₂)]
(hs_m : measurable_set s)
(hs : ∀ (t : set α), measurable_set (s ∩ t) ↔ measurable_set (s ∩ t)) :
measure_theory.condexp m μ f =ᵐ[μ.restrict s] measure_theory.condexp m₂ μ f
If the restriction to a 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].