File: Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean

full_name: MeasureTheory.rnDeriv_ae_eq_condexp
code: theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
    (hf : Integrable f μ) :
    SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] 
goal: MeasureTheory.SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m]

full_name: MeasureTheory.eLpNorm_one_condexp_le_eLpNorm
code: theorem eLpNorm_one_condexp_le_eLpNorm (f : α → ℝ) : eLpNorm (μ[f|m]) 1 μ ≤ eLpNorm f 1 μ 
goal: MeasureTheory.eLpNorm (μ[f|m]) 1 μ ≤ MeasureTheory.eLpNorm f 1 μ

full_name: MeasureTheory.integral_abs_condexp_le
code: theorem integral_abs_condexp_le (f : α → ℝ) : ∫ x, |(μ[f|m]) x| ∂μ ≤ ∫ x, |f x| ∂μ 
goal: ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

full_name: MeasureTheory.setIntegral_abs_condexp_le
code: theorem setIntegral_abs_condexp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
    ∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ 
goal: ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ

full_name: MeasureTheory.ae_bdd_condexp_of_ae_bdd
code: /-- If the real valued function `f` is bounded almost everywhere by `R`, then so is its conditional
expectation. -/
theorem ae_bdd_condexp_of_ae_bdd {R : ℝ≥0} {f : α → ℝ} (hbdd : ∀ᵐ x ∂μ, |f x| ≤ R) :
    ∀ᵐ x ∂μ, |(μ[f|m]) x| ≤ R 
goal: ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

full_name: MeasureTheory.Integrable.uniformIntegrable_condexp
code: /-- Given an integrable function `g`, the conditional expectations of `g` with respect to
a sequence of sub-σ-algebras is uniformly integrable. -/
theorem Integrable.uniformIntegrable_condexp {ι : Type*} [IsFiniteMeasure μ] {g : α → ℝ}
    (hint : Integrable g μ) {ℱ : ι → MeasurableSpace α} (hℱ : ∀ i, ℱ i ≤ m0) :
    UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ 
goal: MeasureTheory.UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ

full_name: MeasureTheory.condexp_stronglyMeasurable_simpleFunc_mul
code: /-- Auxiliary lemma for `condexp_stronglyMeasurable_mul`. -/
theorem condexp_stronglyMeasurable_simpleFunc_mul (hm : m ≤ m0) (f : @SimpleFunc α m ℝ) {g : α → ℝ}
    (hg : Integrable g μ) : μ[(f * g : α → ℝ)|m] =ᵐ[μ] f * μ[g|m] 
goal: μ[↑f * g|m] =ᵐ[μ] ↑f * μ[g|m]

full_name: MeasureTheory.condexp_stronglyMeasurable_mul_of_bound
code: theorem condexp_stronglyMeasurable_mul_of_bound (hm : m ≤ m0) [IsFiniteMeasure μ] {f g : α → ℝ}
    (hf : StronglyMeasurable[m] f) (hg : Integrable g μ) (c : ℝ) (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
    μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]

full_name: MeasureTheory.condexp_stronglyMeasurable_mul_of_bound₀
code: theorem condexp_stronglyMeasurable_mul_of_bound₀ (hm : m ≤ m0) [IsFiniteMeasure μ] {f g : α → ℝ}
    (hf : AEStronglyMeasurable' m f μ) (hg : Integrable g μ) (c : ℝ)
    (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]

full_name: MeasureTheory.condexp_stronglyMeasurable_mul
code: /-- Pull-out property of the conditional expectation. -/
theorem condexp_stronglyMeasurable_mul {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
    (hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]

full_name: MeasureTheory.condexp_stronglyMeasurable_mul₀
code: /-- Pull-out property of the conditional expectation. -/
theorem condexp_stronglyMeasurable_mul₀ {f g : α → ℝ} (hf : AEStronglyMeasurable' m f μ)
    (hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]