ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.lintegral_nnnorm_condexpL2_le
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 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (f : ↥(MeasureTheory.Lp ℝ 2 μ))
Goal: ∫⁻ (x : α) in s, ↑‖↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) f) x‖₊ ∂μ ≤ ∫⁻ (x : α) in s, ↑‖↑↑f x‖₊ ∂μ
-- MeasureTheory.condexpL2_ae_eq_zero_of_ae_eq_zero
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) 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) {f : ↥(MeasureTheory.Lp ℝ 2 μ)} (hf : ↑↑f =ᵐ[μ.restrict s] 0)
Goal: ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) f) =ᵐ[μ.restrict s] 0
-- MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le_real
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) 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {hm : m ≤ m0} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: ∫⁻ (a : α) in t, ↑‖↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ ≤ μ (s ∩ t)
-- MeasureTheory.condexpL2_const_inner
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⟫ 
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)) (c : E)
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
-- MeasureTheory.integral_condexpL2_eq
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 
Arguments: {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E' 2 μ)) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤)
Goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ
-- MeasureTheory.condexpL2_comp_continuousLinearMap
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') 
Arguments: {α : Type u_1} {E' : Type u_3} (𝕜 : Type u_7) [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E'' : Type u_8} (𝕜' : Type u_9) [RCLike 𝕜'] [NormedAddCommGroup E''] [InnerProductSpace 𝕜' E''] [CompleteSpace E''] [NormedSpace ℝ E''] (hm : m ≤ m0) (T : E' →L[ℝ] E'') (f : ↥(MeasureTheory.Lp E' 2 μ))
Goal: ↑↑↑((MeasureTheory.condexpL2 E'' 𝕜' hm) (T.compLp f)) =ᵐ[μ] ↑↑(T.compLp ↑((MeasureTheory.condexpL2 E' 𝕜 hm) f))
-- MeasureTheory.condexpL2_indicator_ae_eq_smul
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 
Arguments: {α : Type u_1} {E' : Type u_3} (𝕜 : Type u_7) [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E')
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
-- MeasureTheory.condexpL2_indicator_eq_toSpanSingleton_comp
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)) 
Arguments: {α : Type u_1} {E' : Type u_3} (𝕜 : Type u_7) [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E')
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))
-- MeasureTheory.setLIntegral_nnnorm_condexpL2_indicator_le
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‖₊ 
Arguments: {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E') {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: ∫⁻ (a : α) in t, ↑‖↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤
  μ (s ∩ t) * ↑‖x‖₊
-- MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le
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‖₊ 
Arguments: {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E') [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: ∫⁻ (a : α), ↑‖↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊
-- MeasureTheory.integrable_condexpL2_indicator
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 
Arguments: {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E')
Goal: MeasureTheory.Integrable (↑↑↑((MeasureTheory.condexpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x))) μ
-- MeasureTheory.aeStronglyMeasurable'_condexpIndSMul
theorem aeStronglyMeasurable'_condexpIndSMul (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) : AEStronglyMeasurable' m (condexpIndSMul hm hs hμs 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) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: MeasureTheory.AEStronglyMeasurable' m (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) μ
-- MeasureTheory.condexpIndSMul_add
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 
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} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x y : G)
Goal: MeasureTheory.condexpIndSMul hm hs hμs (x + y) =
  MeasureTheory.condexpIndSMul hm hs hμs x + MeasureTheory.condexpIndSMul hm hs hμs y
-- MeasureTheory.condexpIndSMul_smul
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 
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} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : ℝ) (x : G)
Goal: MeasureTheory.condexpIndSMul hm hs hμs (c • x) = c • MeasureTheory.condexpIndSMul hm hs hμs x
-- MeasureTheory.condexpIndSMul_smul'
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 
Arguments: {α : Type u_1} {F : Type u_4} {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} [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : 𝕜) (x : F)
Goal: MeasureTheory.condexpIndSMul hm hs hμs (c • x) = c • MeasureTheory.condexpIndSMul hm hs hμs x
-- MeasureTheory.condexpIndSMul_ae_eq_smul
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 
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) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) =ᵐ[μ] fun a =>
  ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1)) a • x
-- MeasureTheory.setLIntegral_nnnorm_condexpIndSMul_le
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‖₊ 
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) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G) {t : Set α} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤)
Goal: ∫⁻ (a : α) in t, ↑‖↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ (s ∩ t) * ↑‖x‖₊
-- MeasureTheory.lintegral_nnnorm_condexpIndSMul_le
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‖₊ 
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) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G) [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: ∫⁻ (a : α), ↑‖↑↑(MeasureTheory.condexpIndSMul hm hs hμs x) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊
-- MeasureTheory.integrable_condexpIndSMul
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 
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.Integrable (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) μ
-- MeasureTheory.condexpIndSMul_empty
theorem condexpIndSMul_empty {x : G} : condexpIndSMul hm MeasurableSet.empty
    ((measure_empty (μ := μ)).le.trans_lt ENNReal.coe_lt_top).ne x = 0 
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} {x : G}
Goal: MeasureTheory.condexpIndSMul hm ⋯ ⋯ x = 0
-- MeasureTheory.setIntegral_condexpL2_indicator
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 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {hm : m ≤ m0} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤)
Goal: ∫ (x : α) in s, ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 ht hμt 1)) x ∂μ =
  (μ (t ∩ s)).toReal
-- MeasureTheory.setIntegral_condexpIndSMul
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 
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} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (x : G')
Goal: ∫ (a : α) in s, ↑↑(MeasureTheory.condexpIndSMul hm ht hμt x) a ∂μ = (μ (t ∩ s)).toReal • x
-- MeasureTheory.condexpL2_indicator_nonneg
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) 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: 0 ≤ᵐ[μ] ↑↑↑((MeasureTheory.condexpL2 ℝ ℝ hm) (MeasureTheory.indicatorConstLp 2 hs hμs 1))
-- MeasureTheory.condexpIndSMul_nonneg
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 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} {E : Type u_10} [NormedLatticeAddCommGroup E] [NormedSpace ℝ E] [OrderedSMul ℝ E] [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E) (hx : 0 ≤ x)
Goal: 0 ≤ᵐ[μ] ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)
-- MeasureTheory.Measure.le_dirac_apply
theorem le_dirac_apply {a} : s.indicator 1 a ≤ dirac a s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} {a : α}
Goal: s.indicator 1 a ≤ (MeasureTheory.Measure.dirac a) s
-- MeasureTheory.Measure.dirac_apply'
@[simp]
theorem dirac_apply' (a : α) (hs : MeasurableSet s) : dirac a s = s.indicator 1 a 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} (a : α) (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.dirac a) s = s.indicator 1 a
-- MeasureTheory.Measure.dirac_apply_of_mem
@[simp]
theorem dirac_apply_of_mem {a : α} (h : a ∈ s) : dirac a s = 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} {a : α} (h : a ∈ s)
Goal: (MeasureTheory.Measure.dirac a) s = 1
-- MeasureTheory.Measure.dirac_apply
@[simp]
theorem dirac_apply [MeasurableSingletonClass α] (a : α) (s : Set α) :
    dirac a s = s.indicator 1 a 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) (s : Set α)
Goal: (MeasureTheory.Measure.dirac a) s = s.indicator 1 a
-- MeasureTheory.Measure.map_dirac
theorem map_dirac {f : α → β} (hf : Measurable f) (a : α) : (dirac a).map f = dirac (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (hf : Measurable f) (a : α)
Goal: MeasureTheory.Measure.map f (MeasureTheory.Measure.dirac a) = MeasureTheory.Measure.dirac (f a)
-- MeasureTheory.Measure.map_const
lemma map_const (μ : Measure α) (c : β) : μ.map (fun _ ↦ c) = (μ Set.univ) • dirac c 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (μ : MeasureTheory.Measure α) (c : β)
Goal: MeasureTheory.Measure.map (fun x => c) μ = μ Set.univ • MeasureTheory.Measure.dirac c
-- MeasureTheory.Measure.restrict_singleton
@[simp]
theorem restrict_singleton (μ : Measure α) (a : α) : μ.restrict {a} = μ {a} • dirac a 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (a : α)
Goal: μ.restrict {a} = μ {a} • MeasureTheory.Measure.dirac a
-- MeasureTheory.Measure.map_eq_sum
theorem map_eq_sum [Countable β] [MeasurableSingletonClass β] (μ : Measure α) (f : α → β)
    (hf : Measurable f) : μ.map f = sum fun b : β => μ (f ⁻¹' {b}) • dirac b 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [Countable β] [MeasurableSingletonClass β] (μ : MeasureTheory.Measure α) (f : α → β) (hf : Measurable f)
Goal: MeasureTheory.Measure.map f μ = MeasureTheory.Measure.sum fun b => μ (f ⁻¹' {b}) • MeasureTheory.Measure.dirac b
-- MeasureTheory.Measure.sum_smul_dirac
@[simp]
theorem sum_smul_dirac [Countable α] [MeasurableSingletonClass α] (μ : Measure α) :
    (sum fun a => μ {a} • dirac a) = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] [Countable α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.sum fun a => μ {a} • MeasureTheory.Measure.dirac a) = μ
-- MeasureTheory.Measure.tsum_indicator_apply_singleton
theorem tsum_indicator_apply_singleton [Countable α] [MeasurableSingletonClass α] (μ : Measure α)
    (s : Set α) (hs : MeasurableSet s) : (∑' x : α, s.indicator (fun x => μ {x}) x) = μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] [Countable α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (s : Set α) (hs : MeasurableSet s)
Goal: ∑' (x : α), s.indicator (fun x => μ {x}) x = μ s
-- MeasureTheory.mem_ae_dirac_iff
theorem mem_ae_dirac_iff {a : α} (hs : MeasurableSet s) : s ∈ ae (dirac a) ↔ a ∈ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} {a : α} (hs : MeasurableSet s)
Goal: s ∈ MeasureTheory.ae (MeasureTheory.Measure.dirac a) ↔ a ∈ s
-- MeasureTheory.ae_dirac_iff
theorem ae_dirac_iff {a : α} {p : α → Prop} (hp : MeasurableSet { x | p x }) :
    (∀ᵐ x ∂dirac a, p x) ↔ p a 
Arguments: {α : Type u_1} [MeasurableSpace α] {a : α} {p : α → Prop} (hp : MeasurableSet {x | p x})
Goal: (∀ᵐ (x : α) ∂MeasureTheory.Measure.dirac a, p x) ↔ p a
-- MeasureTheory.ae_dirac_eq
@[simp]
theorem ae_dirac_eq [MeasurableSingletonClass α] (a : α) : ae (dirac a) = pure a 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α)
Goal: MeasureTheory.ae (MeasureTheory.Measure.dirac a) = pure a
-- MeasureTheory.ae_eq_dirac'
theorem ae_eq_dirac' [MeasurableSingletonClass β] {a : α} {f : α → β} (hf : Measurable f) :
    f =ᵐ[dirac a] const α (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β] {a : α} {f : α → β} (hf : Measurable f)
Goal: f =ᵐ[MeasureTheory.Measure.dirac a] Function.const α (f a)
-- MeasureTheory.ae_eq_dirac
theorem ae_eq_dirac [MeasurableSingletonClass α] {a : α} (f : α → δ) :
    f =ᵐ[dirac a] const α (f a) 
Arguments: {α : Type u_1} {δ : Type u_3} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α} (f : α → δ)
Goal: f =ᵐ[MeasureTheory.Measure.dirac a] Function.const α (f a)
-- MeasureTheory.restrict_dirac'
theorem restrict_dirac' (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    (Measure.dirac a).restrict s = if a ∈ s then Measure.dirac a else 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} {a : α} (hs : MeasurableSet s) [Decidable (a ∈ s)]
Goal: (MeasureTheory.Measure.dirac a).restrict s = if a ∈ s then MeasureTheory.Measure.dirac a else 0
-- MeasureTheory.restrict_dirac
theorem restrict_dirac [MeasurableSingletonClass α] [Decidable (a ∈ s)] :
    (Measure.dirac a).restrict s = if a ∈ s then Measure.dirac a else 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} {a : α} [MeasurableSingletonClass α] [Decidable (a ∈ s)]
Goal: (MeasureTheory.Measure.dirac a).restrict s = if a ∈ s then MeasureTheory.Measure.dirac a else 0
-- MeasureTheory.mutuallySingular_dirac
lemma mutuallySingular_dirac [MeasurableSingletonClass α] (x : α) (μ : Measure α) [NoAtoms μ] :
    Measure.dirac x ⟂ₘ μ 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (x : α) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: (MeasureTheory.Measure.dirac x).MutuallySingular μ
-- MeasureTheory.dirac_eq_dirac_iff_forall_mem_iff_mem
lemma dirac_eq_dirac_iff_forall_mem_iff_mem {x y : α} :
    Measure.dirac x = Measure.dirac y ↔ ∀ A, MeasurableSet A → (x ∈ A ↔ y ∈ A) 
Arguments: {α : Type u_1} [MeasurableSpace α] {x y : α}
Goal: MeasureTheory.Measure.dirac x = MeasureTheory.Measure.dirac y ↔ ∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)
-- MeasureTheory.dirac_ne_dirac_iff_exists_measurableSet
lemma dirac_ne_dirac_iff_exists_measurableSet {x y : α} :
    Measure.dirac x ≠ Measure.dirac y ↔ ∃ A, MeasurableSet A ∧ x ∈ A ∧ y ∉ A 
Arguments: {α : Type u_1} [MeasurableSpace α] {x y : α}
Goal: MeasureTheory.Measure.dirac x ≠ MeasureTheory.Measure.dirac y ↔ ∃ A, MeasurableSet A ∧ x ∈ A ∧ y ∉ A
-- MeasureTheory.dirac_ne_dirac
lemma dirac_ne_dirac [SeparatesPoints α] {x y : α} (x_ne_y : x ≠ y) :
    Measure.dirac x ≠ Measure.dirac y 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.SeparatesPoints α] {x y : α} (x_ne_y : x ≠ y)
Goal: MeasureTheory.Measure.dirac x ≠ MeasureTheory.Measure.dirac y
-- MeasureTheory.dirac_ne_dirac_iff
lemma dirac_ne_dirac_iff [SeparatesPoints α] {x y : α} :
    Measure.dirac x ≠ Measure.dirac y ↔ x ≠ y 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.SeparatesPoints α] {x y : α}
Goal: MeasureTheory.Measure.dirac x ≠ MeasureTheory.Measure.dirac y ↔ x ≠ y
-- MeasureTheory.dirac_eq_dirac_iff
lemma dirac_eq_dirac_iff [SeparatesPoints α] {x y : α} :
    Measure.dirac x = Measure.dirac y ↔ x = y 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.SeparatesPoints α] {x y : α}
Goal: MeasureTheory.Measure.dirac x = MeasureTheory.Measure.dirac y ↔ x = y
-- MeasureTheory.injective_dirac
lemma injective_dirac [SeparatesPoints α] :
    Function.Injective (fun (x : α) ↦ dirac x) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSpace.SeparatesPoints α]
Goal: Function.Injective fun x => MeasureTheory.Measure.dirac x
-- Besicovitch.SatelliteConfig.h
structure Besicovitch.SatelliteConfig (α : Type*) [MetricSpace α] (N : ℕ) (τ : ℝ) where
  c : Fin N.succ → α
  r : Fin N.succ → ℝ
  rpos : ∀ i, 0 < r i
  h : Pairwise fun i j =>
    r i ≤ dist (c i) (c j) ∧ r j ≤ τ * r i ∨ r j ≤ dist (c j) (c i) ∧ r i ≤ τ * r j
  hlast : ∀ i < last N, r i ≤ dist (c i) (c (last N)) ∧ r (last N) ≤ τ * r i
  inter : ∀ i < last N, dist (c i) (c (last N)) ≤ r i + r (last N)
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (self : Besicovitch.SatelliteConfig α N τ)
Goal: Pairwise fun i j =>
  self.r i ≤ dist (self.c i) (self.c j) ∧ self.r j ≤ τ * self.r i ∨
    self.r j ≤ dist (self.c j) (self.c i) ∧ self.r i ≤ τ * self.r j
-- Besicovitch.SatelliteConfig.hlast
structure Besicovitch.SatelliteConfig (α : Type*) [MetricSpace α] (N : ℕ) (τ : ℝ) where
  c : Fin N.succ → α
  r : Fin N.succ → ℝ
  rpos : ∀ i, 0 < r i
  h : Pairwise fun i j =>
    r i ≤ dist (c i) (c j) ∧ r j ≤ τ * r i ∨ r j ≤ dist (c j) (c i) ∧ r i ≤ τ * r j
  hlast : ∀ i < last N, r i ≤ dist (c i) (c (last N)) ∧ r (last N) ≤ τ * r i
  inter : ∀ i < last N, dist (c i) (c (last N)) ≤ r i + r (last N)
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (self : Besicovitch.SatelliteConfig α N τ) (i : Fin (N + 1))
Goal: i < Fin.last N → self.r i ≤ dist (self.c i) (self.c (Fin.last N)) ∧ self.r (Fin.last N) ≤ τ * self.r i