ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left'
theorem setToL1SCLM_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ hT'' f = setToL1SCLM α E μ hT f + setToL1SCLM α E μ hT' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hT'' : MeasureTheory.DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T'' s = T s + T' s) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT'') f =
  (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f + (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left
theorem setToL1SCLM_smul_left (c : ℝ) (hT : DominatedFinMeasAdditive μ T C) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ (hT.smul c) f = c • setToL1SCLM α E μ hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T : Set α → E →L[ℝ] F} {C : ℝ} (c : ℝ) (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ ⋯) f = c • (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left'
theorem setToL1SCLM_smul_left' (c : ℝ) (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C')
    (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ hT' f = c • setToL1SCLM α E μ hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T T' : Set α → E →L[ℝ] F} {C C' : ℝ} (c : ℝ) (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T' s = c • T s) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f = c • (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f
-- MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le
theorem norm_setToL1SCLM_le {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hC : 0 ≤ C) : ‖setToL1SCLM α E μ hT‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C)
Goal: ‖MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT‖ ≤ C
-- MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le'
theorem norm_setToL1SCLM_le' {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C) :
    ‖setToL1SCLM α E μ hT‖ ≤ max C 0 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C)
Goal: ‖MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT‖ ≤ max C 0
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_const
theorem setToL1SCLM_const [IsFiniteMeasure μ] {T : Set α → E →L[ℝ] F} {C : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (x : E) :
    setToL1SCLM α E μ hT (simpleFunc.indicatorConst 1 MeasurableSet.univ (measure_ne_top μ _) x) =
      T univ x 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (x : E)
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) (MeasureTheory.Lp.simpleFunc.indicatorConst 1 ⋯ ⋯ x) = (T Set.univ) x
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left
theorem setToL1SCLM_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ hT f ≤ setToL1SCLM α E μ hT' f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α) (x : E), (T s) x ≤ (T' s) x) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f ≤ (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left'
theorem setToL1SCLM_mono_left' {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ hT f ≤ setToL1SCLM α E μ hT' f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f ≤ (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_nonneg
theorem setToL1SCLM_nonneg {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁ₛ[μ] G'}
    (hf : 0 ≤ f) : 0 ≤ setToL1SCLM α G' μ hT f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) {f : ↥(MeasureTheory.Lp.simpleFunc G' 1 μ)} (hf : 0 ≤ f)
Goal: 0 ≤ (MeasureTheory.L1.SimpleFunc.setToL1SCLM α G' μ hT) f
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono
theorem setToL1SCLM_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁ₛ[μ] G'}
    (hfg : f ≤ g) : setToL1SCLM α G' μ hT f ≤ setToL1SCLM α G' μ hT g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) {f g : ↥(MeasureTheory.Lp.simpleFunc G' 1 μ)} (hfg : f ≤ g)
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α G' μ hT) f ≤ (MeasureTheory.L1.SimpleFunc.setToL1SCLM α G' μ hT) g
-- MeasureTheory.L1.setToL1_eq_setToL1SCLM
theorem setToL1_eq_setToL1SCLM (hT : DominatedFinMeasAdditive μ T C) (f : α →₁ₛ[μ] E) :
    setToL1 hT f = setToL1SCLM α E μ hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) ↑f = (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f
-- MeasureTheory.L1.setToL1_eq_setToL1'
theorem setToL1_eq_setToL1' (hT : DominatedFinMeasAdditive μ T C)
    (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1' 𝕜 hT h_smul f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f = (MeasureTheory.L1.setToL1' 𝕜 hT h_smul) f
-- MeasureTheory.L1.setToL1_zero_left
@[simp]
theorem setToL1_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C)
    (f : α →₁[μ] E) : setToL1 hT f = 0 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ 0 C) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f = 0
-- MeasureTheory.L1.setToL1_zero_left'
theorem setToL1_zero_left' (hT : DominatedFinMeasAdditive μ T C)
    (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁[μ] E) : setToL1 hT f = 0 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f = 0
-- MeasureTheory.L1.setToL1_congr_left
theorem setToL1_congr_left (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T')
    (f : α →₁[μ] E) : setToL1 hT f = setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] (T T' : Set α → E →L[ℝ] F) {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (h : T = T') (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f = (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_congr_left'
theorem setToL1_congr_left' (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁[μ] E) :
    setToL1 hT f = setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] (T T' : Set α → E →L[ℝ] F) {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f = (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_add_left
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
    setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T T' : Set α → E →L[ℝ] F} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 ⋯) f = (MeasureTheory.L1.setToL1 hT) f + (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_add_left'
theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
    setToL1 hT'' f = setToL1 hT f + setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hT'' : MeasureTheory.DominatedFinMeasAdditive μ T'' C'') (h_add : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T'' s = T s + T' s) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT'') f = (MeasureTheory.L1.setToL1 hT) f + (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_smul_left
theorem setToL1_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α →₁[μ] E) :
    setToL1 (hT.smul c) f = c • setToL1 hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (c : ℝ) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 ⋯) f = c • (MeasureTheory.L1.setToL1 hT) f
-- MeasureTheory.L1.setToL1_smul_left'
theorem setToL1_smul_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (c : ℝ)
    (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁[μ] E) :
    setToL1 hT' f = c • setToL1 hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T T' : Set α → E →L[ℝ] F} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (c : ℝ) (h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T' s = c • T s) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT') f = c • (MeasureTheory.L1.setToL1 hT) f
-- MeasureTheory.L1.setToL1_smul
theorem setToL1_smul (hT : DominatedFinMeasAdditive μ T C)
    (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) (f : α →₁[μ] E) :
    setToL1 hT (c • f) = c • setToL1 hT f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) (c : 𝕜) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) (c • f) = c • (MeasureTheory.L1.setToL1 hT) f
-- MeasureTheory.L1.setToL1_simpleFunc_indicatorConst
theorem setToL1_simpleFunc_indicatorConst (hT : DominatedFinMeasAdditive μ T C) {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s < ∞) (x : E) :
    setToL1 hT (simpleFunc.indicatorConst 1 hs hμs.ne x) = T s x 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤) (x : E)
Goal: (MeasureTheory.L1.setToL1 hT) ↑(MeasureTheory.Lp.simpleFunc.indicatorConst 1 hs ⋯ x) = (T s) x
-- MeasureTheory.L1.setToL1_indicatorConstLp
theorem setToL1_indicatorConstLp (hT : DominatedFinMeasAdditive μ T C) {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) :
    setToL1 hT (indicatorConstLp 1 hs hμs x) = T s x 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E)
Goal: (MeasureTheory.L1.setToL1 hT) (MeasureTheory.indicatorConstLp 1 hs hμs x) = (T s) x
-- MeasureTheory.L1.setToL1_const
theorem setToL1_const [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) :
    setToL1 hT (indicatorConstLp 1 MeasurableSet.univ (measure_ne_top _ _) x) = T univ x 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} [MeasureTheory.IsFiniteMeasure μ] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (x : E)
Goal: (MeasureTheory.L1.setToL1 hT) (MeasureTheory.indicatorConstLp 1 ⋯ ⋯ x) = (T Set.univ) x
-- MeasureTheory.L1.setToL1_mono_left'
theorem setToL1_mono_left' {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁[μ] E) :
    setToL1 hT f ≤ setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f ≤ (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_mono_left
theorem setToL1_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁[μ] E) : setToL1 hT f ≤ setToL1 hT' f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (hTT' : ∀ (s : Set α) (x : E), (T s) x ≤ (T' s) x) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: (MeasureTheory.L1.setToL1 hT) f ≤ (MeasureTheory.L1.setToL1 hT') f
-- MeasureTheory.L1.setToL1_nonneg
theorem setToL1_nonneg {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁[μ] G'}
    (hf : 0 ≤ f) : 0 ≤ setToL1 hT f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) {f : ↥(MeasureTheory.Lp G' 1 μ)} (hf : 0 ≤ f)
Goal: 0 ≤ (MeasureTheory.L1.setToL1 hT) f
-- MeasureTheory.L1.setToL1_mono
theorem setToL1_mono {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : DominatedFinMeasAdditive μ T C)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁[μ] G'}
    (hfg : f ≤ g) : setToL1 hT f ≤ setToL1 hT g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [CompleteSpace G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) {f g : ↥(MeasureTheory.Lp G' 1 μ)} (hfg : f ≤ g)
Goal: (MeasureTheory.L1.setToL1 hT) f ≤ (MeasureTheory.L1.setToL1 hT) g
-- MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM
theorem norm_setToL1_le_norm_setToL1SCLM (hT : DominatedFinMeasAdditive μ T C) :
    ‖setToL1 hT‖ ≤ ‖setToL1SCLM α E μ hT‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C)
Goal: ‖MeasureTheory.L1.setToL1 hT‖ ≤ ‖MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT‖
-- MeasureTheory.L1.norm_setToL1_le_mul_norm
theorem norm_setToL1_le_mul_norm (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C)
    (f : α →₁[μ] E) : ‖setToL1 hT f‖ ≤ C * ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: ‖(MeasureTheory.L1.setToL1 hT) f‖ ≤ C * ‖f‖
-- MeasureTheory.L1.norm_setToL1_le_mul_norm'
theorem norm_setToL1_le_mul_norm' (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) :
    ‖setToL1 hT f‖ ≤ max C 0 * ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: ‖(MeasureTheory.L1.setToL1 hT) f‖ ≤ max C 0 * ‖f‖
-- MeasureTheory.L1.norm_setToL1_le
theorem norm_setToL1_le (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : ‖setToL1 hT‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C)
Goal: ‖MeasureTheory.L1.setToL1 hT‖ ≤ C
-- MeasureTheory.L1.norm_setToL1_le'
theorem norm_setToL1_le' (hT : DominatedFinMeasAdditive μ T C) : ‖setToL1 hT‖ ≤ max C 0 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C)
Goal: ‖MeasureTheory.L1.setToL1 hT‖ ≤ max C 0
-- MeasureTheory.L1.setToL1_lipschitz
theorem setToL1_lipschitz (hT : DominatedFinMeasAdditive μ T C) :
    LipschitzWith (Real.toNNReal C) (setToL1 hT) 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C)
Goal: LipschitzWith C.toNNReal ⇑(MeasureTheory.L1.setToL1 hT)