File: Mathlib/Probability/Kernel/Condexp.lean

full_name: MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id
code: theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id [TopologicalSpace F]
    (hm : m ≤ mΩ) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x : Ω × Ω => f x.2)
      (@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) 
goal: MeasureTheory.AEStronglyMeasurable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (id ω, id ω)) μ)

full_name: MeasureTheory.Integrable.comp_snd_map_prod_id
code: theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_id [NormedAddCommGroup F] (hm : m ≤ mΩ)
    (hf : Integrable f μ) : Integrable (fun x : Ω × Ω => f x.2)
      (@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) 
goal: MeasureTheory.Integrable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (id ω, id ω)) μ)

full_name: ProbabilityTheory.condexpKernel_def
code: /-- Kernel associated with the conditional expectation with respect to a σ-algebra. It satisfies
`μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condexpKernel μ m ω)`.
It is defined as the conditional distribution of the identity given the identity, where the second
identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` with σ-algebra `m ⊓ mΩ`.
We use `m ⊓ mΩ` instead of `m` to ensure that it is a sub-σ-algebra of `mΩ`. We then use
`Kernel.comap` to get a kernel from `m` to `mΩ` instead of from `m ⊓ mΩ` to `mΩ`. -/
noncomputable irreducible_def condexpKernel (μ : Measure Ω) [IsFiniteMeasure μ]
    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ 
goal: ProbabilityTheory.condexpKernel μ m = (ProbabilityTheory.condDistrib id id μ).comap id ⋯

full_name: ProbabilityTheory.definition._@.Mathlib.Probability.Kernel.Condexp._hyg.296
code: /-- Kernel associated with the conditional expectation with respect to a σ-algebra. It satisfies
`μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condexpKernel μ m ω)`.
It is defined as the conditional distribution of the identity given the identity, where the second
identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` with σ-algebra `m ⊓ mΩ`.
We use `m ⊓ mΩ` instead of `m` to ensure that it is a sub-σ-algebra of `mΩ`. We then use
`Kernel.comap` to get a kernel from `m` to `mΩ` instead of from `m ⊓ mΩ` to `mΩ`. -/
noncomputable irreducible_def condexpKernel (μ : Measure Ω) [IsFiniteMeasure μ]
    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ 
goal: ProbabilityTheory.Kernel Ω Ω

full_name: ProbabilityTheory.wrapped._@.Mathlib.Probability.Kernel.Condexp._hyg.296
code: /-- Kernel associated with the conditional expectation with respect to a σ-algebra. It satisfies
`μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condexpKernel μ m ω)`.
It is defined as the conditional distribution of the identity given the identity, where the second
identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` with σ-algebra `m ⊓ mΩ`.
We use `m ⊓ mΩ` instead of `m` to ensure that it is a sub-σ-algebra of `mΩ`. We then use
`Kernel.comap` to get a kernel from `m` to `mΩ` instead of from `m ⊓ mΩ` to `mΩ`. -/
noncomputable irreducible_def condexpKernel (μ : Measure Ω) [IsFiniteMeasure μ]
    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ 
goal: Subtype (Eq @ProbabilityTheory.definition✝)

full_name: ProbabilityTheory.condexpKernel
code: /-- Kernel associated with the conditional expectation with respect to a σ-algebra. It satisfies
`μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condexpKernel μ m ω)`.
It is defined as the conditional distribution of the identity given the identity, where the second
identity is understood as a map from `Ω` with the σ-algebra `mΩ` to `Ω` with σ-algebra `m ⊓ mΩ`.
We use `m ⊓ mΩ` instead of `m` to ensure that it is a sub-σ-algebra of `mΩ`. We then use
`Kernel.comap` to get a kernel from `m` to `mΩ` instead of from `m ⊓ mΩ` to `mΩ`. -/
noncomputable irreducible_def condexpKernel (μ : Measure Ω) [IsFiniteMeasure μ]
    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ 
goal: ProbabilityTheory.Kernel Ω Ω

full_name: ProbabilityTheory.condexpKernel_apply_eq_condDistrib
code: lemma condexpKernel_apply_eq_condDistrib {ω : Ω} :
    condexpKernel μ m ω = @condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _ (id ω) 
goal: (ProbabilityTheory.condexpKernel μ m) ω = (ProbabilityTheory.condDistrib id id μ) (id ω)

full_name: ProbabilityTheory.instIsMarkovKernelCondexpKernel
code: instance : IsMarkovKernel (condexpKernel μ m) 
goal: ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.condexpKernel μ m)

full_name: ProbabilityTheory.measurable_condexpKernel
code: theorem measurable_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    Measurable[m] fun ω => condexpKernel μ m ω s 
goal: Measurable fun ω => ((ProbabilityTheory.condexpKernel μ m) ω) s

full_name: ProbabilityTheory.stronglyMeasurable_condexpKernel
code: theorem stronglyMeasurable_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    StronglyMeasurable[m] fun ω => condexpKernel μ m ω s 
goal: MeasureTheory.StronglyMeasurable fun ω => ((ProbabilityTheory.condexpKernel μ m) ω) s

full_name: MeasureTheory.AEStronglyMeasurable.integral_condexpKernel
code: theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condexpKernel [NormedSpace ℝ F]
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
goal: MeasureTheory.AEStronglyMeasurable (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ

full_name: ProbabilityTheory.aestronglyMeasurable'_integral_condexpKernel
code: theorem aestronglyMeasurable'_integral_condexpKernel [NormedSpace ℝ F]
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable' m (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
goal: MeasureTheory.AEStronglyMeasurable' m (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ

full_name: MeasureTheory.Integrable.condexpKernel_ae
code: theorem _root_.MeasureTheory.Integrable.condexpKernel_ae (hf_int : Integrable f μ) :
    ∀ᵐ ω ∂μ, Integrable f (condexpKernel μ m ω) 
goal: ∀ᵐ (ω : Ω) ∂μ, MeasureTheory.Integrable f ((ProbabilityTheory.condexpKernel μ m) ω)

full_name: MeasureTheory.Integrable.integral_norm_condexpKernel
code: theorem _root_.MeasureTheory.Integrable.integral_norm_condexpKernel (hf_int : Integrable f μ) :
    Integrable (fun ω => ∫ y, ‖f y‖ ∂condexpKernel μ m ω) μ 
goal: MeasureTheory.Integrable (fun ω => ∫ (y : Ω), ‖f y‖ ∂(ProbabilityTheory.condexpKernel μ m) ω) μ

full_name: MeasureTheory.Integrable.norm_integral_condexpKernel
code: theorem _root_.MeasureTheory.Integrable.norm_integral_condexpKernel [NormedSpace ℝ F]
    (hf_int : Integrable f μ) :
    Integrable (fun ω => ‖∫ y, f y ∂condexpKernel μ m ω‖) μ 
goal: MeasureTheory.Integrable (fun ω => ‖∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω‖) μ

full_name: MeasureTheory.Integrable.integral_condexpKernel
code: theorem _root_.MeasureTheory.Integrable.integral_condexpKernel [NormedSpace ℝ F]
    (hf_int : Integrable f μ) :
    Integrable (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
goal: MeasureTheory.Integrable (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ

full_name: ProbabilityTheory.integrable_toReal_condexpKernel
code: theorem integrable_toReal_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    Integrable (fun ω => (condexpKernel μ m ω s).toReal) μ 
goal: MeasureTheory.Integrable (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) μ

full_name: ProbabilityTheory.condexpKernel_ae_eq_condexp'
code: lemma condexpKernel_ae_eq_condexp' [IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ] μ⟦s | m ⊓ mΩ⟧ 
goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω => 1|m ⊓ mΩ]

full_name: ProbabilityTheory.condexpKernel_ae_eq_condexp
code: lemma condexpKernel_ae_eq_condexp [IsFiniteMeasure μ]
    (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ] μ⟦s | m⟧ 
goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω => 1|m]

full_name: ProbabilityTheory.condexpKernel_ae_eq_trim_condexp
code: lemma condexpKernel_ae_eq_trim_condexp [IsFiniteMeasure μ]
    (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ.trim hm] μ⟦s | m⟧ 
goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ.trim hm] μ[s.indicator fun ω => 1|m]

full_name: ProbabilityTheory.condexp_ae_eq_integral_condexpKernel'
code: theorem condexp_ae_eq_integral_condexpKernel' [NormedAddCommGroup F] {f : Ω → F}
    [NormedSpace ℝ F] [CompleteSpace F] (hf_int : Integrable f μ) :
    μ[f|m ⊓ mΩ] =ᵐ[μ] fun ω => ∫ y, f y ∂condexpKernel μ m ω 
goal: μ[f|m ⊓ mΩ] =ᵐ[μ] fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω

full_name: ProbabilityTheory.condexp_ae_eq_integral_condexpKernel
code: /-- The conditional expectation of `f` with respect to a σ-algebra `m` is almost everywhere equal to
the integral `∫ y, f y ∂(condexpKernel μ m ω)`. -/
theorem condexp_ae_eq_integral_condexpKernel [NormedAddCommGroup F] {f : Ω → F}
    [NormedSpace ℝ F] [CompleteSpace F] (hm : m ≤ mΩ) (hf_int : Integrable f μ) :
    μ[f|m] =ᵐ[μ] fun ω => ∫ y, f y ∂condexpKernel μ m ω 
goal: μ[f|m] =ᵐ[μ] fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω