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

full_name: MeasureTheory.lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero
code: theorem lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') 
goal: ↑↑↑f =ᵐ[μ] 0

full_name: MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero'
code: theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 
goal: ↑↑f =ᵐ[μ] 0

full_name: MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq'
code: /-- **Uniqueness of the conditional expectation** -/
theorem Lp.ae_eq_of_forall_setIntegral_eq' (hm : m ≤ m0) (f g : Lp E' p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
    (hfg : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hf_meas : AEStronglyMeasurable' m f μ) (hg_meas : AEStronglyMeasurable' m g μ) :
    f =ᵐ[μ] g 
goal: ↑↑f =ᵐ[μ] ↑↑g

full_name: MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite'
code: theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    {f g : α → F'} (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hfm : AEStronglyMeasurable' m f μ) (hgm : AEStronglyMeasurable' m g μ) : f =ᵐ[μ] g 
goal: f =ᵐ[μ] g

full_name: MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eq
code: /-- Let `m` be a sub-σ-algebra of `m0`, `f` an `m0`-measurable function and `g` an `m`-measurable
function, such that their integrals coincide on `m`-measurable sets with finite measure.
Then `∫ x in s, ‖g x‖ ∂μ ≤ ∫ x in s, ‖f x‖ ∂μ` on all `m`-measurable sets with finite measure. -/
theorem integral_norm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
    (hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
    (hgi : IntegrableOn g s μ)
    (hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫ x in s, ‖g x‖ ∂μ) ≤ ∫ x in s, ‖f x‖ ∂μ 
goal: ∫ (x : α) in s, ‖g x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ

full_name: MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq
code: /-- Let `m` be a sub-σ-algebra of `m0`, `f` an `m0`-measurable function and `g` an `m`-measurable
function, such that their integrals coincide on `m`-measurable sets with finite measure.
Then `∫⁻ x in s, ‖g x‖₊ ∂μ ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ` on all `m`-measurable sets with finite
measure. -/
theorem lintegral_nnnorm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
    (hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
    (hgi : IntegrableOn g s μ)
    (hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫⁻ x in s, ‖g x‖₊ ∂μ) ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ 
goal: ∫⁻ (x : α) in s, ↑‖g x‖₊ ∂μ ≤ ∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ