Conditional expectation #
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We build the conditional expectation of an integrable function f with value in a Banach space
with respect to a measure μ (defined on a measurable space structure m0) and a measurable space
structure m with hm : m ≤ m0 (a sub-sigma-algebra). This is an m-strongly measurable
function μ[f|hm] which is integrable and verifies ∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ
for all m-measurable sets s. It is unique as an element of L¹.
The construction is done in four steps:
- Define the conditional expectation of an
L²function, as an element ofL². This is the orthogonal projection on the subspace of almost everywherem-measurable functions. - Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map
set α → (E →L[ℝ] (α →₁[μ] E))which to a set associates a linear map. That linear map sendsx ∈ Eto the conditional expectation of the indicator of the set with valuex. - Extend that map to
condexp_L1_clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E). This is done using the same construction as the Bochner integral (see the filemeasure_theory/integral/set_to_L1). - Define the conditional expectation of a function
f : α → E, which is an integrable functionα → Eequal to 0 iffis not integrable, and equal to anm-measurable representative ofcondexp_L1_clmapplied to[f], the equivalence class offinL¹.
The first step is done in measure_theory.function.conditional_expectation.condexp_L2, the two
next steps in measure_theory.function.conditional_expectation.condexp_L1 and the final step is
performed in this file.
Main results #
The conditional expectation and its properties
condexp (m : measurable_space α) (μ : measure α) (f : α → E): conditional expectation offwith respect tom.integrable_condexp:condexpis integrable.strongly_measurable_condexp:condexpism-strongly-measurable.set_integral_condexp (hf : integrable f μ) (hs : measurable_set[m] s): ifm ≤ m0(the σ-algebra over which the measure is defined), then the conditional expectation verifies∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μfor anym-measurable sets.
While condexp is function-valued, we also define condexp_L1 with value in L1 and a continuous
linear map condexp_L1_clm from L1 to L1. condexp should be used in most cases.
Uniqueness of the conditional expectation
ae_eq_condexp_of_forall_set_integral_eq: an a.e.m-measurable function which verifies the equality of integrals is a.e. equal tocondexp.
Notations #
For a measure μ defined on a measurable space structure m0, another measurable space structure
m with hm : m ≤ m0 (a sub-σ-algebra) and a function f, we define the notation
μ[f|m] = condexp m μ f.
Tags #
conditional expectation, conditional expected value
@[irreducible]
noncomputable
def
measure_theory.condexp
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
(m : measurable_space α)
{m0 : measurable_space α}
(μ : measure_theory.measure α)
(f : α → F') :
α → F'
Conditional expectation of a function. It is defined as 0 if any one of the following conditions is true:
mis not a sub-σ-algebra ofm0,μis not σ-finite with respect tom,fis not integrable.
Equations
- measure_theory.condexp m μ f = dite (m ≤ m0) (λ (hm : m ≤ m0), dite (measure_theory.sigma_finite (μ.trim hm) ∧ measure_theory.integrable f μ) (λ (h : measure_theory.sigma_finite (μ.trim hm) ∧ measure_theory.integrable f μ), ite (measure_theory.strongly_measurable f) f (measure_theory.ae_strongly_measurable'.mk ⇑(measure_theory.condexp_L1 hm μ f) _)) (λ (h : ¬(measure_theory.sigma_finite (μ.trim hm) ∧ measure_theory.integrable f μ)), 0)) (λ (hm : ¬m ≤ m0), 0)
theorem
measure_theory.condexp_of_not_le
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hm_not : ¬m ≤ m0) :
measure_theory.condexp m μ f = 0
theorem
measure_theory.condexp_of_not_sigma_finite
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hm : m ≤ m0)
(hμm_not : ¬measure_theory.sigma_finite (μ.trim hm)) :
measure_theory.condexp m μ f = 0
theorem
measure_theory.condexp_of_sigma_finite
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hm : m ≤ m0)
[hμm : measure_theory.sigma_finite (μ.trim hm)] :
theorem
measure_theory.condexp_of_strongly_measurable
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[hμm : measure_theory.sigma_finite (μ.trim hm)]
{f : α → F'}
(hf : measure_theory.strongly_measurable f)
(hfi : measure_theory.integrable f μ) :
measure_theory.condexp m μ f = f
theorem
measure_theory.condexp_const
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
(c : F')
[measure_theory.is_finite_measure μ] :
measure_theory.condexp m μ (λ (x : α), c) = λ (_x : α), c
theorem
measure_theory.condexp_ae_eq_condexp_L1
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[hμm : measure_theory.sigma_finite (μ.trim hm)]
(f : α → F') :
measure_theory.condexp m μ f =ᵐ[μ] ⇑(measure_theory.condexp_L1 hm μ f)
theorem
measure_theory.condexp_ae_eq_condexp_L1_clm
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(hf : measure_theory.integrable f μ) :
measure_theory.condexp m μ f =ᵐ[μ] ⇑(⇑(measure_theory.condexp_L1_clm hm μ) (measure_theory.integrable.to_L1 f hf))
theorem
measure_theory.condexp_undef
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hf : ¬measure_theory.integrable f μ) :
measure_theory.condexp m μ f = 0
@[simp]
theorem
measure_theory.condexp_zero
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α} :
measure_theory.condexp m μ 0 = 0
theorem
measure_theory.strongly_measurable_condexp
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'} :
theorem
measure_theory.condexp_congr_ae
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → F'}
(h : f =ᵐ[μ] g) :
measure_theory.condexp m μ f =ᵐ[μ] measure_theory.condexp m μ g
theorem
measure_theory.condexp_of_ae_strongly_measurable'
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[hμm : measure_theory.sigma_finite (μ.trim hm)]
{f : α → F'}
(hf : measure_theory.ae_strongly_measurable' m f μ)
(hfi : measure_theory.integrable f μ) :
measure_theory.condexp m μ f =ᵐ[μ] f
theorem
measure_theory.integrable_condexp
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'} :
theorem
measure_theory.set_integral_condexp
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
{s : set α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(hf : measure_theory.integrable f μ)
(hs : measurable_set s) :
The integral of the conditional expectation μ[f|hm] over an m-measurable set is equal to
the integral of f on that set.
theorem
measure_theory.integral_condexp
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f : α → F'}
(hm : m ≤ m0)
[hμm : measure_theory.sigma_finite (μ.trim hm)]
(hf : measure_theory.integrable f μ) :
theorem
measure_theory.ae_eq_condexp_of_forall_set_integral_eq
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
{f g : α → F'}
(hf : measure_theory.integrable f μ)
(hg_int_finite : ∀ (s : set α), measurable_set s → ⇑μ s < ⊤ → measure_theory.integrable_on g s μ)
(hg_eq : ∀ (s : set α), measurable_set s → ⇑μ s < ⊤ → ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, f x ∂μ)
(hgm : measure_theory.ae_strongly_measurable' m g μ) :
g =ᵐ[μ] measure_theory.condexp m μ f
Uniqueness of the conditional expectation
If a function is a.e. m-measurable, verifies an integrability condition and has same integral
as f on all m-measurable sets, then it is a.e. equal to μ[f|hm].
theorem
measure_theory.condexp_bot'
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[hμ : μ.ae.ne_bot]
(f : α → F') :
theorem
measure_theory.condexp_bot_ae_eq
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m0 : measurable_space α}
{μ : measure_theory.measure α}
(f : α → F') :
theorem
measure_theory.condexp_bot
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m0 : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_probability_measure μ]
(f : α → F') :
theorem
measure_theory.condexp_add
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → F'}
(hf : measure_theory.integrable f μ)
(hg : measure_theory.integrable g μ) :
measure_theory.condexp m μ (f + g) =ᵐ[μ] measure_theory.condexp m μ f + measure_theory.condexp m μ g
theorem
measure_theory.condexp_finset_sum
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{ι : Type u_2}
{s : finset ι}
{f : ι → α → F'}
(hf : ∀ (i : ι), i ∈ s → measure_theory.integrable (f i) μ) :
measure_theory.condexp m μ (s.sum (λ (i : ι), f i)) =ᵐ[μ] s.sum (λ (i : ι), measure_theory.condexp m μ (f i))
theorem
measure_theory.condexp_smul
{α : Type u_1}
{F' : Type u_3}
{𝕜 : Type u_4}
[is_R_or_C 𝕜]
[normed_add_comm_group F']
[normed_space 𝕜 F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(c : 𝕜)
(f : α → F') :
measure_theory.condexp m μ (c • f) =ᵐ[μ] c • measure_theory.condexp m μ f
theorem
measure_theory.condexp_neg
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(f : α → F') :
measure_theory.condexp m μ (-f) =ᵐ[μ] -measure_theory.condexp m μ f
theorem
measure_theory.condexp_sub
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → F'}
(hf : measure_theory.integrable f μ)
(hg : measure_theory.integrable g μ) :
measure_theory.condexp m μ (f - g) =ᵐ[μ] measure_theory.condexp m μ f - measure_theory.condexp m μ g
theorem
measure_theory.condexp_condexp_of_le
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{f : α → F'}
{m₁ m₂ m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm₁₂ : m₁ ≤ m₂)
(hm₂ : m₂ ≤ m0)
[measure_theory.sigma_finite (μ.trim hm₂)] :
measure_theory.condexp m₁ μ (measure_theory.condexp m₂ μ f) =ᵐ[μ] measure_theory.condexp m₁ μ f
theorem
measure_theory.condexp_mono
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_2}
[normed_lattice_add_comm_group E]
[complete_space E]
[normed_space ℝ E]
[ordered_smul ℝ E]
{f g : α → E}
(hf : measure_theory.integrable f μ)
(hg : measure_theory.integrable g μ)
(hfg : f ≤ᵐ[μ] g) :
measure_theory.condexp m μ f ≤ᵐ[μ] measure_theory.condexp m μ g
theorem
measure_theory.condexp_nonneg
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_2}
[normed_lattice_add_comm_group E]
[complete_space E]
[normed_space ℝ E]
[ordered_smul ℝ E]
{f : α → E}
(hf : 0 ≤ᵐ[μ] f) :
0 ≤ᵐ[μ] measure_theory.condexp m μ f
theorem
measure_theory.condexp_nonpos
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_2}
[normed_lattice_add_comm_group E]
[complete_space E]
[normed_space ℝ E]
[ordered_smul ℝ E]
{f : α → E}
(hf : f ≤ᵐ[μ] 0) :
measure_theory.condexp m μ f ≤ᵐ[μ] 0
theorem
measure_theory.tendsto_condexp_L1_of_dominated_convergence
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
{fs : ℕ → α → F'}
{f : α → F'}
(bound_fs : α → ℝ)
(hfs_meas : ∀ (n : ℕ), measure_theory.ae_strongly_measurable (fs n) μ)
(h_int_bound_fs : measure_theory.integrable bound_fs μ)
(hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hfs : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), fs n x) filter.at_top (nhds (f x))) :
filter.tendsto (λ (n : ℕ), measure_theory.condexp_L1 hm μ (fs n)) filter.at_top (nhds (measure_theory.condexp_L1 hm μ f))
Lebesgue dominated convergence theorem: sufficient conditions under which almost
everywhere convergence of a sequence of functions implies the convergence of their image by
condexp_L1.
theorem
measure_theory.tendsto_condexp_unique
{α : Type u_1}
{F' : Type u_3}
[normed_add_comm_group F']
[normed_space ℝ F']
[complete_space F']
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(fs gs : ℕ → α → F')
(f g : α → F')
(hfs_int : ∀ (n : ℕ), measure_theory.integrable (fs n) μ)
(hgs_int : ∀ (n : ℕ), measure_theory.integrable (gs n) μ)
(hfs : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), fs n x) filter.at_top (nhds (f x)))
(hgs : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), gs n x) filter.at_top (nhds (g x)))
(bound_fs : α → ℝ)
(h_int_bound_fs : measure_theory.integrable bound_fs μ)
(bound_gs : α → ℝ)
(h_int_bound_gs : measure_theory.integrable bound_gs μ)
(hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x)
(hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x)
(hfg : ∀ (n : ℕ), measure_theory.condexp m μ (fs n) =ᵐ[μ] measure_theory.condexp m μ (gs n)) :
measure_theory.condexp m μ f =ᵐ[μ] measure_theory.condexp m μ g
If two sequences of functions have a.e. equal conditional expectations at each step, converge and verify dominated convergence hypotheses, then the conditional expectations of their limits are a.e. equal.