File: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_ConditionalExpectation_CondexpL2___macroRules_MeasureTheory_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_ConditionalExpectation_CondexpL2___macroRules_MeasureTheory_term⟪_,_⟫₂_1»
code: local notation "⟪" x ", " y "⟫₂" => @inner 𝕜 (α →₂[μ] E) _ x y
goal: Lean.Macro

full_name: MeasureTheory.condexpL2
code: /-- Conditional expectation of a function in L2 with respect to a sigma-algebra -/
noncomputable def condexpL2 (hm : m ≤ m0) : (α →₂[μ] E) →L[𝕜] lpMeas E 𝕜 m 2 μ 
goal: ↥(MeasureTheory.Lp E 2 μ) →L[𝕜] ↥(MeasureTheory.lpMeas E 𝕜 m 2 μ)

full_name: MeasureTheory.aeStronglyMeasurable'_condexpL2
code: theorem aeStronglyMeasurable'_condexpL2 (hm : m ≤ m0) (f : α →₂[μ] E) :
    AEStronglyMeasurable' (β := E) m (condexpL2 E 𝕜 hm f) μ 
goal: MeasureTheory.AEStronglyMeasurable' m (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) μ

full_name: MeasureTheory.integrableOn_condexpL2_of_measure_ne_top
code: theorem integrableOn_condexpL2_of_measure_ne_top (hm : m ≤ m0) (hμs : μ s ≠ ∞) (f : α →₂[μ] E) :
    IntegrableOn (E := E) (condexpL2 E 𝕜 hm f) s μ 
goal: MeasureTheory.IntegrableOn (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) s μ

full_name: MeasureTheory.integrable_condexpL2_of_isFiniteMeasure
code: theorem integrable_condexpL2_of_isFiniteMeasure (hm : m ≤ m0) [IsFiniteMeasure μ] {f : α →₂[μ] E} :
    Integrable (β := E) (condexpL2 E 𝕜 hm f) μ 
goal: MeasureTheory.Integrable (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) μ

full_name: MeasureTheory.norm_condexpL2_le_one
code: theorem norm_condexpL2_le_one (hm : m ≤ m0) : ‖@condexpL2 α E 𝕜 _ _ _ _ _ _ μ hm‖ ≤ 1 
goal: ‖MeasureTheory.condexpL2 E 𝕜 hm‖ ≤ 1

full_name: MeasureTheory.norm_condexpL2_le
code: theorem norm_condexpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) : ‖condexpL2 E 𝕜 hm f‖ ≤ ‖f‖ 
goal: ‖(MeasureTheory.condexpL2 E 𝕜 hm) f‖ ≤ ‖f‖

full_name: MeasureTheory.eLpNorm_condexpL2_le
code: theorem eLpNorm_condexpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) :
    eLpNorm (F := E) (condexpL2 E 𝕜 hm f) 2 μ ≤ eLpNorm f 2 μ 
goal: MeasureTheory.eLpNorm (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) 2 μ ≤ MeasureTheory.eLpNorm (↑↑f) 2 μ

full_name: MeasureTheory.norm_condexpL2_coe_le
code: theorem norm_condexpL2_coe_le (hm : m ≤ m0) (f : α →₂[μ] E) :
    ‖(condexpL2 E 𝕜 hm f : α →₂[μ] E)‖ ≤ ‖f‖ 
goal: ‖↑((MeasureTheory.condexpL2 E 𝕜 hm) f)‖ ≤ ‖f‖

full_name: MeasureTheory.inner_condexpL2_left_eq_right
code: theorem inner_condexpL2_left_eq_right (hm : m ≤ m0) {f g : α →₂[μ] E} :
    ⟪(condexpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, (condexpL2 E 𝕜 hm g : α →₂[μ] E)⟫₂ 
goal: ⟪↑((MeasureTheory.condexpL2 E 𝕜 hm) f), g⟫_𝕜 = ⟪f, ↑((MeasureTheory.condexpL2 E 𝕜 hm) g)⟫_𝕜

full_name: MeasureTheory.condexpL2_indicator_of_measurable
code: theorem condexpL2_indicator_of_measurable (hm : m ≤ m0) (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞)
    (c : E) :
    (condexpL2 E 𝕜 hm (indicatorConstLp 2 (hm s hs) hμs c) : α →₂[μ] E) =
      indicatorConstLp 2 (hm s hs) hμs c 
goal: ↑((MeasureTheory.condexpL2 E 𝕜 hm) (MeasureTheory.indicatorConstLp 2 ⋯ hμs c)) =
  MeasureTheory.indicatorConstLp 2 ⋯ hμs c

full_name: MeasureTheory.inner_condexpL2_eq_inner_fun
code: theorem inner_condexpL2_eq_inner_fun (hm : m ≤ m0) (f g : α →₂[μ] E)
    (hg : AEStronglyMeasurable' m g μ) :
    ⟪(condexpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, g⟫₂ 
goal: ⟪↑((MeasureTheory.condexpL2 E 𝕜 hm) f), g⟫_𝕜 = ⟪f, g⟫_𝕜

full_name: MeasureTheory.integral_condexpL2_eq_of_fin_meas_real
code: theorem integral_condexpL2_eq_of_fin_meas_real (f : Lp 𝕜 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 𝕜 𝕜 hm f : α → 𝕜) x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 𝕜 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

full_name: MeasureTheory.lintegral_nnnorm_condexpL2_le
code: theorem lintegral_nnnorm_condexpL2_le (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) (f : Lp ℝ 2 μ) :
    ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ 
goal: ∫⁻ (x : α) in s, ↑‖↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ ≤ ∫⁻ (x : α) in s, ↑‖↑↑f x‖₊ ∂μ

full_name: MeasureTheory.condexpL2_ae_eq_zero_of_ae_eq_zero
code: theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
    (hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) 
goal: ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) f) =ᵐ[μ.restrict s] 0

full_name: MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le_real
code: theorem lintegral_nnnorm_condexpL2_indicator_le_real (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    ∫⁻ a in t, ‖(condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ ≤ μ (s ∩ t) 
goal: ∫⁻ (a : α) in t, ↑‖↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ ≤ μ (s ∩ t)

full_name: MeasureTheory.condexpL2_const_inner
code: /-- `condexpL2` commutes with taking inner products with constants. See the lemma
`condexpL2_comp_continuousLinearMap` for a more general result about commuting with continuous
linear maps. -/
theorem condexpL2_const_inner (hm : m ≤ m0) (f : Lp E 2 μ) (c : E) :
    condexpL2 𝕜 𝕜 hm (((Lp.memℒp f).const_inner c).toLp fun a => ⟪c, f a⟫) =ᵐ[μ]
    fun a => ⟪c, (condexpL2 E 𝕜 hm f : α → E) a⟫ 
goal: ↑↑↑((MeasureTheory.condexpL2 𝕜 𝕜 hm) (MeasureTheory.Memℒp.toLp (fun a => ⟪c, ↑↑f a⟫_𝕜) ⋯)) =ᵐ[μ] fun a =>
  ⟪c, ↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f) a⟫_𝕜

full_name: MeasureTheory.integral_condexpL2_eq
code: /-- `condexpL2` verifies the equality of integrals defining the conditional expectation. -/
theorem integral_condexpL2_eq (hm : m ≤ m0) (f : Lp E' 2 μ) (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, (condexpL2 E' 𝕜 hm f : α → E') x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

full_name: MeasureTheory.condexpL2_comp_continuousLinearMap
code: theorem condexpL2_comp_continuousLinearMap (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : α →₂[μ] E') :
    (condexpL2 E'' 𝕜' hm (T.compLp f) : α →₂[μ] E'') =ᵐ[μ]
    T.compLp (condexpL2 E' 𝕜 hm f : α →₂[μ] E') 
goal: ↑↑↑((MeasureTheory.condexpL2 E'' 𝕜' hm) (T.compLp f)) =ᵐ[μ] ↑↑(T.compLp ↑((MeasureTheory.condexpL2 E' 𝕜 hm) f))

full_name: MeasureTheory.condexpL2_indicator_ae_eq_smul
code: theorem condexpL2_indicator_ae_eq_smul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : E') :
    condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) =ᵐ[μ] fun a =>
      (condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs (1 : ℝ)) : α → ℝ) a • x 
goal: ↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) =ᵐ[μ] fun a =>
  ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1)) a • x

full_name: MeasureTheory.condexpL2_indicator_eq_toSpanSingleton_comp
code: theorem condexpL2_indicator_eq_toSpanSingleton_comp (hm : m ≤ m0) (hs : MeasurableSet s)
    (hμs : μ s ≠ ∞) (x : E') : (condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) : α →₂[μ] E') =
    (toSpanSingleton ℝ x).compLp (condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1)) 
goal: ↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) =
  (ContinuousLinearMap.toSpanSingleton ℝ x).compLp
    ↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1))

full_name: MeasureTheory.setLIntegral_nnnorm_condexpL2_indicator_le
code: theorem setLIntegral_nnnorm_condexpL2_indicator_le (hm : m ≤ m0) (hs : MeasurableSet s)
    (hμs : μ s ≠ ∞) (x : E') {t : Set α} (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    ∫⁻ a in t, ‖(condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) : α → E') a‖₊ ∂μ ≤
    μ (s ∩ t) * ‖x‖₊ 
goal: ∫⁻ (a : α) in t, ↑‖↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤
  μ (s ∩ t) * ↑‖x‖₊

full_name: MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le
code: theorem lintegral_nnnorm_condexpL2_indicator_le (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : E') [SigmaFinite (μ.trim hm)] :
    ∫⁻ a, ‖(condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x) : α → E') a‖₊ ∂μ ≤ μ s * ‖x‖₊ 
goal: ∫⁻ (a : α), ↑‖↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊

full_name: MeasureTheory.integrable_condexpL2_indicator
code: /-- If the measure `μ.trim hm` is sigma-finite, then the conditional expectation of a measurable set
with finite measure is integrable. -/
theorem integrable_condexpL2_indicator (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E') :
    Integrable (β := E') (condexpL2 E' 𝕜 hm (indicatorConstLp 2 hs hμs x)) μ 
goal: MeasureTheory.Integrable (↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x))) μ

full_name: MeasureTheory.condexpIndSMul
code: /-- Conditional expectation of the indicator of a measurable set with finite measure, in L2. -/
noncomputable def condexpIndSMul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    Lp G 2 μ 
goal: ↥(MeasureTheory.Lp G 2 μ)

full_name: MeasureTheory.aeStronglyMeasurable'_condexpIndSMul
code: theorem aeStronglyMeasurable'_condexpIndSMul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) : AEStronglyMeasurable' m (condexpIndSMul hm hs hμs x) μ 
goal: MeasureTheory.AEStronglyMeasurable' m (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) μ

full_name: MeasureTheory.condexpIndSMul_add
code: theorem condexpIndSMul_add (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x y : G) :
    condexpIndSMul hm hs hμs (x + y) = condexpIndSMul hm hs hμs x + condexpIndSMul hm hs hμs y 
goal: MeasureTheory.condexpIndSMul hm hs hμs (x + y) =
  MeasureTheory.condexpIndSMul hm hs hμs x + MeasureTheory.condexpIndSMul hm hs hμs y

full_name: MeasureTheory.condexpIndSMul_smul
code: theorem condexpIndSMul_smul (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : ℝ) (x : G) :
    condexpIndSMul hm hs hμs (c • x) = c • condexpIndSMul hm hs hμs x 
goal: MeasureTheory.condexpIndSMul hm hs hμs (c • x) = c • MeasureTheory.condexpIndSMul hm hs hμs x

full_name: MeasureTheory.condexpIndSMul_smul'
code: theorem condexpIndSMul_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s)
    (hμs : μ s ≠ ∞) (c : 𝕜) (x : F) :
    condexpIndSMul hm hs hμs (c • x) = c • condexpIndSMul hm hs hμs x 
goal: MeasureTheory.condexpIndSMul hm hs hμs (c • x) = c • MeasureTheory.condexpIndSMul hm hs hμs x

full_name: MeasureTheory.condexpIndSMul_ae_eq_smul
code: theorem condexpIndSMul_ae_eq_smul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    condexpIndSMul hm hs hμs x =ᵐ[μ] fun a =>
      (condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a • x 
goal: ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) =ᵐ[μ] fun a =>
  ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1)) a • x

full_name: MeasureTheory.setLIntegral_nnnorm_condexpIndSMul_le
code: theorem setLIntegral_nnnorm_condexpIndSMul_le (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) {t : Set α} (ht : MeasurableSet[m] t) (hμt : μ t ≠ ∞) :
    (∫⁻ a in t, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ) ≤ μ (s ∩ t) * ‖x‖₊ 
goal: ∫⁻ (a : α) in t, ↑‖↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ (s ∩ t) * ↑‖x‖₊

full_name: MeasureTheory.lintegral_nnnorm_condexpIndSMul_le
code: theorem lintegral_nnnorm_condexpIndSMul_le (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) [SigmaFinite (μ.trim hm)] : ∫⁻ a, ‖condexpIndSMul hm hs hμs x a‖₊ ∂μ ≤ μ s * ‖x‖₊ 
goal: ∫⁻ (a : α), ↑‖↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊

full_name: MeasureTheory.integrable_condexpIndSMul
code: /-- If the measure `μ.trim hm` is sigma-finite, then the conditional expectation of a measurable set
with finite measure is integrable. -/
theorem integrable_condexpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hs : MeasurableSet s)
    (hμs : μ s ≠ ∞) (x : G) : Integrable (condexpIndSMul hm hs hμs x) μ 
goal: MeasureTheory.Integrable (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) μ

full_name: MeasureTheory.condexpIndSMul_empty
code: theorem condexpIndSMul_empty {x : G} : condexpIndSMul hm MeasurableSet.empty
    ((measure_empty (μ := μ)).le.trans_lt ENNReal.coe_lt_top).ne x = 0 
goal: MeasureTheory.condexpIndSMul hm ⋯ ⋯ x = 0

full_name: MeasureTheory.setIntegral_condexpL2_indicator
code: theorem setIntegral_condexpL2_indicator (hs : MeasurableSet[m] s) (ht : MeasurableSet t)
    (hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) :
    ∫ x in s, (condexpL2 ℝ ℝ hm (indicatorConstLp 2 ht hμt 1) : α → ℝ) x ∂μ = (μ (t ∩ s)).toReal 
goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 ht hμt 1)) x ∂μ =
  (μ (t ∩ s)).toReal

full_name: MeasureTheory.setIntegral_condexpIndSMul
code: theorem setIntegral_condexpIndSMul (hs : MeasurableSet[m] s) (ht : MeasurableSet t)
    (hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) (x : G') :
    ∫ a in s, (condexpIndSMul hm ht hμt x) a ∂μ = (μ (t ∩ s)).toReal • x 
goal: ∫ (a : α) in s, ↑↑(MeasureTheory.condexpIndSMul hm ht hμt x) a ∂μ = (μ (t ∩ s)).toReal • x

full_name: MeasureTheory.condexpL2_indicator_nonneg
code: theorem condexpL2_indicator_nonneg (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    [SigmaFinite (μ.trim hm)] : (0 : α → ℝ) ≤ᵐ[μ]
    condexpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) 
goal: 0 ≤ᵐ[μ] ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1))

full_name: MeasureTheory.condexpIndSMul_nonneg
code: theorem condexpIndSMul_nonneg {E} [NormedLatticeAddCommGroup E] [NormedSpace ℝ E] [OrderedSMul ℝ E]
    [SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) (hx : 0 ≤ x) :
    (0 : α → E) ≤ᵐ[μ] condexpIndSMul hm hs hμs x 
goal: 0 ≤ᵐ[μ] ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)