ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.condexpIndL1_disjoint_union
theorem condexpIndL1_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1 hm μ (s ∪ t) x = condexpIndL1 hm μ s x + condexpIndL1 hm μ t x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : s ∩ t = ∅) (x : G)
Goal: MeasureTheory.condexpIndL1 hm μ (s ∪ t) x = MeasureTheory.condexpIndL1 hm μ s x + MeasureTheory.condexpIndL1 hm μ t x
-- MeasureTheory.condexpInd_ae_eq_condexpIndSMul
theorem condexpInd_ae_eq_condexpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    condexpInd G hm μ s x =ᵐ[μ] condexpIndSMul hm hs hμs x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: ↑↑((MeasureTheory.condexpInd G hm μ s) x) =ᵐ[μ] ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)
-- MeasureTheory.aestronglyMeasurable'_condexpInd
theorem aestronglyMeasurable'_condexpInd (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    AEStronglyMeasurable' m (condexpInd G hm μ s x) μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑((MeasureTheory.condexpInd G hm μ s) x)) μ
-- MeasureTheory.condexpInd_empty
@[simp]
theorem condexpInd_empty : condexpInd G hm μ ∅ = (0 : G →L[ℝ] α →₁[μ] G) 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: MeasureTheory.condexpInd G hm μ ∅ = 0
-- MeasureTheory.condexpInd_smul'
theorem condexpInd_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (x : F) :
    condexpInd F hm μ s (c • x) = c • condexpInd F hm μ s x 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (x : F)
Goal: (MeasureTheory.condexpInd F hm μ s) (c • x) = c • (MeasureTheory.condexpInd F hm μ s) x
-- MeasureTheory.norm_condexpInd_apply_le
theorem norm_condexpInd_apply_le (x : G) : ‖condexpInd G hm μ s x‖ ≤ (μ s).toReal * ‖x‖ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (x : G)
Goal: ‖(MeasureTheory.condexpInd G hm μ s) x‖ ≤ (μ s).toReal * ‖x‖
-- MeasureTheory.norm_condexpInd_le
theorem norm_condexpInd_le : ‖(condexpInd G hm μ s : G →L[ℝ] α →₁[μ] G)‖ ≤ (μ s).toReal 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: ‖MeasureTheory.condexpInd G hm μ s‖ ≤ (μ s).toReal
-- MeasureTheory.condexpInd_disjoint_union_apply
theorem condexpInd_disjoint_union_apply (hs : MeasurableSet s) (ht : MeasurableSet t)
    (hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpInd G hm μ (s ∪ t) x = condexpInd G hm μ s x + condexpInd G hm μ t x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : s ∩ t = ∅) (x : G)
Goal: (MeasureTheory.condexpInd G hm μ (s ∪ t)) x =
  (MeasureTheory.condexpInd G hm μ s) x + (MeasureTheory.condexpInd G hm μ t) x
-- MeasureTheory.condexpInd_disjoint_union
theorem condexpInd_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) : (condexpInd G hm μ (s ∪ t) : G →L[ℝ] α →₁[μ] G) =
    condexpInd G hm μ s + condexpInd G hm μ t 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : s ∩ t = ∅)
Goal: MeasureTheory.condexpInd G hm μ (s ∪ t) = MeasureTheory.condexpInd G hm μ s + MeasureTheory.condexpInd G hm μ t
-- MeasureTheory.dominatedFinMeasAdditive_condexpInd
theorem dominatedFinMeasAdditive_condexpInd (hm : m ≤ m0) (μ : Measure α)
    [SigmaFinite (μ.trim hm)] :
    DominatedFinMeasAdditive μ (condexpInd G hm μ : Set α → G →L[ℝ] α →₁[μ] G) 1 
Arguments: {α : Type u_1} (G : Type u_5) [NormedAddCommGroup G] {m m0 : MeasurableSpace α} [NormedSpace ℝ G] (hm : m ≤ m0) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: MeasureTheory.DominatedFinMeasAdditive μ (MeasureTheory.condexpInd G hm μ) 1
-- MeasureTheory.setIntegral_condexpInd
theorem setIntegral_condexpInd (hs : MeasurableSet[m] s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (x : G') : ∫ a in s, condexpInd G' hm μ t x a ∂μ = (μ (t ∩ s)).toReal • x 
Arguments: {α : Type u_1} {G' : Type u_6} [NormedAddCommGroup G'] [NormedSpace ℝ G'] [CompleteSpace G'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (x : G')
Goal: ∫ (a : α) in s, ↑↑((MeasureTheory.condexpInd G' hm μ t) x) a ∂μ = (μ (t ∩ s)).toReal • x
-- MeasureTheory.condexpInd_of_measurable
theorem condexpInd_of_measurable (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) (c : G) :
    condexpInd G hm μ s c = indicatorConstLp 1 (hm s hs) hμs c 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : G)
Goal: (MeasureTheory.condexpInd G hm μ s) c = MeasureTheory.indicatorConstLp 1 ⋯ hμs c
-- MeasureTheory.condexpInd_nonneg
theorem condexpInd_nonneg {E} [NormedLatticeAddCommGroup E] [NormedSpace ℝ E] [OrderedSMul ℝ E]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) (hx : 0 ≤ x) : 0 ≤ condexpInd E hm μ s x 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {E : Type u_8} [NormedLatticeAddCommGroup E] [NormedSpace ℝ E] [OrderedSMul ℝ E] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E) (hx : 0 ≤ x)
Goal: 0 ≤ (MeasureTheory.condexpInd E hm μ s) x
-- MeasureTheory.condexpL1CLM_smul
theorem condexpL1CLM_smul (c : 𝕜) (f : α →₁[μ] F') :
    condexpL1CLM F' hm μ (c • f) = c • condexpL1CLM F' hm μ f 
Arguments: {α : Type u_1} {F' : Type u_4} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (c : 𝕜) (f : ↥(MeasureTheory.Lp F' 1 μ))
Goal: (MeasureTheory.condexpL1CLM F' hm μ) (c • f) = c • (MeasureTheory.condexpL1CLM F' hm μ) f
-- MeasureTheory.condexpL1CLM_indicatorConstLp
theorem condexpL1CLM_indicatorConstLp (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : F') :
    (condexpL1CLM F' hm μ) (indicatorConstLp 1 hs hμs x) = condexpInd F' hm μ s x 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : F')
Goal: (MeasureTheory.condexpL1CLM F' hm μ) (MeasureTheory.indicatorConstLp 1 hs hμs x) =
  (MeasureTheory.condexpInd F' hm μ s) x
-- MeasureTheory.condexpL1CLM_indicatorConst
theorem condexpL1CLM_indicatorConst (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : F') :
    (condexpL1CLM F' hm μ) ↑(simpleFunc.indicatorConst 1 hs hμs x) = condexpInd F' hm μ s x 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : F')
Goal: (MeasureTheory.condexpL1CLM F' hm μ) ↑(MeasureTheory.Lp.simpleFunc.indicatorConst 1 hs hμs x) =
  (MeasureTheory.condexpInd F' hm μ s) x
-- MeasureTheory.setIntegral_condexpL1CLM_of_measure_ne_top
theorem setIntegral_condexpL1CLM_of_measure_ne_top (f : α →₁[μ] F') (hs : MeasurableSet[m] s)
    (hμs : μ s ≠ ∞) : ∫ x in s, condexpL1CLM F' hm μ f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {s : Set α} (f : ↥(MeasureTheory.Lp F' 1 μ)) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤)
Goal: ∫ (x : α) in s, ↑↑((MeasureTheory.condexpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ
-- MeasureTheory.setIntegral_condexpL1CLM
theorem setIntegral_condexpL1CLM (f : α →₁[μ] F') (hs : MeasurableSet[m] s) :
    ∫ x in s, condexpL1CLM F' hm μ f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {s : Set α} (f : ↥(MeasureTheory.Lp F' 1 μ)) (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, ↑↑((MeasureTheory.condexpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ
-- MeasureTheory.aestronglyMeasurable'_condexpL1CLM
theorem aestronglyMeasurable'_condexpL1CLM (f : α →₁[μ] F') :
    AEStronglyMeasurable' m (condexpL1CLM F' hm μ f) μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (f : ↥(MeasureTheory.Lp F' 1 μ))
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑((MeasureTheory.condexpL1CLM F' hm μ) f)) μ
-- MeasureTheory.condexpL1CLM_lpMeas
theorem condexpL1CLM_lpMeas (f : lpMeas F' ℝ m 1 μ) :
    condexpL1CLM F' hm μ (f : α →₁[μ] F') = ↑f 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (f : ↥(MeasureTheory.lpMeas F' ℝ m 1 μ))
Goal: (MeasureTheory.condexpL1CLM F' hm μ) ↑f = ↑f
-- MeasureTheory.condexpL1CLM_of_aestronglyMeasurable'
theorem condexpL1CLM_of_aestronglyMeasurable' (f : α →₁[μ] F') (hfm : AEStronglyMeasurable' m f μ) :
    condexpL1CLM F' hm μ f = f 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (f : ↥(MeasureTheory.Lp F' 1 μ)) (hfm : MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ)
Goal: (MeasureTheory.condexpL1CLM F' hm μ) f = f
-- MeasureTheory.condexpL1_undef
theorem condexpL1_undef (hf : ¬Integrable f μ) : condexpL1 hm μ f = 0 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} (hf : ¬MeasureTheory.Integrable f μ)
Goal: MeasureTheory.condexpL1 hm μ f = 0
-- MeasureTheory.condexpL1_eq
theorem condexpL1_eq (hf : Integrable f μ) : condexpL1 hm μ f = condexpL1CLM F' hm μ (hf.toL1 f) 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.condexpL1 hm μ f = (MeasureTheory.condexpL1CLM F' hm μ) (MeasureTheory.Integrable.toL1 f hf)
-- MeasureTheory.condexpL1_zero
@[simp]
theorem condexpL1_zero : condexpL1 hm μ (0 : α → F') = 0 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: MeasureTheory.condexpL1 hm μ 0 = 0
-- MeasureTheory.condexpL1_measure_zero
@[simp]
theorem condexpL1_measure_zero (hm : m ≤ m0) : condexpL1 hm (0 : Measure α) f = 0 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {f : α → F'} (hm : m ≤ m0)
Goal: MeasureTheory.condexpL1 hm 0 f = 0
-- MeasureTheory.aestronglyMeasurable'_condexpL1
theorem aestronglyMeasurable'_condexpL1 {f : α → F'} :
    AEStronglyMeasurable' m (condexpL1 hm μ f) μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'}
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑(MeasureTheory.condexpL1 hm μ f)) μ
-- MeasureTheory.condexpL1_congr_ae
theorem condexpL1_congr_ae (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (h : f =ᵐ[μ] g) :
    condexpL1 hm μ f = condexpL1 hm μ g 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → F'} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (h : f =ᵐ[μ] g)
Goal: MeasureTheory.condexpL1 hm μ f = MeasureTheory.condexpL1 hm μ g
-- MeasureTheory.integrable_condexpL1
theorem integrable_condexpL1 (f : α → F') : Integrable (condexpL1 hm μ f) μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (f : α → F')
Goal: MeasureTheory.Integrable (↑↑(MeasureTheory.condexpL1 hm μ f)) μ
-- MeasureTheory.setIntegral_condexpL1
theorem setIntegral_condexpL1 (hf : Integrable f μ) (hs : MeasurableSet[m] s) :
    ∫ x in s, condexpL1 hm μ f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} {s : Set α} (hf : MeasureTheory.Integrable f μ) (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, ↑↑(MeasureTheory.condexpL1 hm μ f) x ∂μ = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.condexpL1_add
theorem condexpL1_add (hf : Integrable f μ) (hg : Integrable g μ) :
    condexpL1 hm μ (f + g) = condexpL1 hm μ f + condexpL1 hm μ g 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f g : α → F'} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.condexpL1 hm μ (f + g) = MeasureTheory.condexpL1 hm μ f + MeasureTheory.condexpL1 hm μ g
-- MeasureTheory.condexpL1_neg
theorem condexpL1_neg (f : α → F') : condexpL1 hm μ (-f) = -condexpL1 hm μ f 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (f : α → F')
Goal: MeasureTheory.condexpL1 hm μ (-f) = -MeasureTheory.condexpL1 hm μ f
-- MeasureTheory.condexpL1_smul
theorem condexpL1_smul (c : 𝕜) (f : α → F') : condexpL1 hm μ (c • f) = c • condexpL1 hm μ f 
Arguments: {α : Type u_1} {F' : Type u_4} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (c : 𝕜) (f : α → F')
Goal: MeasureTheory.condexpL1 hm μ (c • f) = c • MeasureTheory.condexpL1 hm μ f
-- MeasureTheory.condexpL1_sub
theorem condexpL1_sub (hf : Integrable f μ) (hg : Integrable g μ) :
    condexpL1 hm μ (f - g) = condexpL1 hm μ f - condexpL1 hm μ g 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f g : α → F'} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.condexpL1 hm μ (f - g) = MeasureTheory.condexpL1 hm μ f - MeasureTheory.condexpL1 hm μ g
-- MeasureTheory.condexpL1_of_aestronglyMeasurable'
theorem condexpL1_of_aestronglyMeasurable' (hfm : AEStronglyMeasurable' m f μ)
    (hfi : Integrable f μ) : condexpL1 hm μ f =ᵐ[μ] f 
Arguments: {α : Type u_1} {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ) (hfi : MeasureTheory.Integrable f μ)
Goal: ↑↑(MeasureTheory.condexpL1 hm μ f) =ᵐ[μ] f
-- MeasureTheory.condexpL1_mono
theorem condexpL1_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    condexpL1 hm μ f ≤ᵐ[μ] condexpL1 hm μ g 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {E : Type u_8} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] [OrderedSMul ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : f ≤ᵐ[μ] g)
Goal: ↑↑(MeasureTheory.condexpL1 hm μ f) ≤ᵐ[μ] ↑↑(MeasureTheory.condexpL1 hm μ g)
-- MeasureTheory.aeStronglyMeasurable'_condexpL2
theorem aeStronglyMeasurable'_condexpL2 (hm : m ≤ m0) (f : α →₂[μ] E) :
    AEStronglyMeasurable' (β := E) m (condexpL2 E 𝕜 hm f) μ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) μ
-- MeasureTheory.integrableOn_condexpL2_of_measure_ne_top
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 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hμs : μ s ≠ ⊤) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.IntegrableOn (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) s μ
-- MeasureTheory.integrable_condexpL2_of_isFiniteMeasure
theorem integrable_condexpL2_of_isFiniteMeasure (hm : m ≤ m0) [IsFiniteMeasure μ] {f : α →₂[μ] E} :
    Integrable (β := E) (condexpL2 E 𝕜 hm f) μ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] {f : ↥(MeasureTheory.Lp E 2 μ)}
Goal: MeasureTheory.Integrable (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) μ
-- MeasureTheory.norm_condexpL2_le_one
theorem norm_condexpL2_le_one (hm : m ≤ m0) : ‖@condexpL2 α E 𝕜 _ _ _ _ _ _ μ hm‖ ≤ 1 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0)
Goal: ‖MeasureTheory.condexpL2 E 𝕜 hm‖ ≤ 1
-- MeasureTheory.norm_condexpL2_le
theorem norm_condexpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) : ‖condexpL2 E 𝕜 hm f‖ ≤ ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: ‖(MeasureTheory.condexpL2 E 𝕜 hm) f‖ ≤ ‖f‖
-- MeasureTheory.eLpNorm_condexpL2_le
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 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.eLpNorm (↑↑↑((MeasureTheory.condexpL2 E 𝕜 hm) f)) 2 μ ≤ MeasureTheory.eLpNorm (↑↑f) 2 μ
-- MeasureTheory.norm_condexpL2_coe_le
theorem norm_condexpL2_coe_le (hm : m ≤ m0) (f : α →₂[μ] E) :
    ‖(condexpL2 E 𝕜 hm f : α →₂[μ] E)‖ ≤ ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: ‖↑((MeasureTheory.condexpL2 E 𝕜 hm) f)‖ ≤ ‖f‖
-- MeasureTheory.inner_condexpL2_left_eq_right
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 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f g : ↥(MeasureTheory.Lp E 2 μ)}
Goal: ⟪↑((MeasureTheory.condexpL2 E 𝕜 hm) f), g⟫_𝕜 = ⟪f, ↑((MeasureTheory.condexpL2 E 𝕜 hm) g)⟫_𝕜
-- MeasureTheory.condexpL2_indicator_of_measurable
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 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E)
Goal: ↑((MeasureTheory.condexpL2 E 𝕜 hm) (MeasureTheory.indicatorConstLp 2 ⋯ hμs c)) =
  MeasureTheory.indicatorConstLp 2 ⋯ hμs c
-- MeasureTheory.inner_condexpL2_eq_inner_fun
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 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f g : ↥(MeasureTheory.Lp E 2 μ)) (hg : MeasureTheory.AEStronglyMeasurable' m (↑↑g) μ)
Goal: ⟪↑((MeasureTheory.condexpL2 E 𝕜 hm) f), g⟫_𝕜 = ⟪f, g⟫_𝕜
-- MeasureTheory.integral_condexpL2_eq_of_fin_meas_real
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 
Arguments: {α : Type u_1} {𝕜 : Type u_7} [RCLike 𝕜] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} (f : ↥(MeasureTheory.Lp 𝕜 2 μ)) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤)
Goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 𝕜 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ