ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm
theorem norm_setToSimpleFunc_le_sum_mul_norm (T : Set α → F →L[ℝ] F') {C : ℝ}
    (hT_norm : ∀ s, MeasurableSet s → ‖T s‖ ≤ C * (μ s).toReal) (f : α →ₛ F) :
    ‖f.setToSimpleFunc T‖ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → F →L[ℝ] F') {C : ℝ} (hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * (μ s).toReal) (f : MeasureTheory.SimpleFunc α F)
Goal: ‖MeasureTheory.SimpleFunc.setToSimpleFunc T f‖ ≤ C * ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal * ‖x‖
-- MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable
theorem norm_setToSimpleFunc_le_sum_mul_norm_of_integrable (T : Set α → E →L[ℝ] F') {C : ℝ}
    (hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) (f : α →ₛ E)
    (hf : Integrable f μ) :
    ‖f.setToSimpleFunc T‖ ≤ C * ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal * ‖x‖ 
Arguments: {α : Type u_1} {E : Type u_2} {F' : Type u_4} [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_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal) (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: ‖MeasureTheory.SimpleFunc.setToSimpleFunc T f‖ ≤ C * ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal * ‖x‖
-- MeasureTheory.SimpleFunc.setToSimpleFunc_indicator
theorem setToSimpleFunc_indicator (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0)
    {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (x : F) :
    SimpleFunc.setToSimpleFunc T
        (SimpleFunc.piecewise s hs (SimpleFunc.const α x) (SimpleFunc.const α 0)) =
      T s x 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0) {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (x : F)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T
    (MeasureTheory.SimpleFunc.piecewise s hs (MeasureTheory.SimpleFunc.const α x)
      (MeasureTheory.SimpleFunc.const α 0)) =
  (T s) x
-- MeasureTheory.SimpleFunc.setToSimpleFunc_const'
theorem setToSimpleFunc_const' [Nonempty α] (T : Set α → F →L[ℝ] F') (x : F)
    {m : MeasurableSpace α} : SimpleFunc.setToSimpleFunc T (SimpleFunc.const α x) = T univ x 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] [Nonempty α] (T : Set α → F →L[ℝ] F') (x : F) {m : MeasurableSpace α}
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (MeasureTheory.SimpleFunc.const α x) = (T Set.univ) x
-- MeasureTheory.SimpleFunc.setToSimpleFunc_const
theorem setToSimpleFunc_const (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0) (x : F)
    {m : MeasurableSpace α} : SimpleFunc.setToSimpleFunc T (SimpleFunc.const α x) = T univ x 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0) (x : F) {m : MeasurableSpace α}
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (MeasureTheory.SimpleFunc.const α x) = (T Set.univ) x
-- MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul
theorem norm_eq_sum_mul (f : α →₁ₛ[μ] G) :
    ‖f‖ = ∑ x ∈ (toSimpleFunc f).range, (μ (toSimpleFunc f ⁻¹' {x})).toReal * ‖x‖ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc G 1 μ))
Goal: ‖f‖ =
  ∑ x ∈ (MeasureTheory.Lp.simpleFunc.toSimpleFunc f).range,
    (μ (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖
-- MeasureTheory.L1.SimpleFunc.setToL1S_eq_setToSimpleFunc
theorem setToL1S_eq_setToSimpleFunc (T : Set α → E →L[ℝ] F) (f : α →₁ₛ[μ] E) :
    setToL1S T f = (toSimpleFunc f).setToSimpleFunc T 
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) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f =
  MeasureTheory.SimpleFunc.setToSimpleFunc T (MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.L1.SimpleFunc.setToL1S_zero_left
@[simp]
theorem setToL1S_zero_left (f : α →₁ₛ[μ] E) : setToL1S (0 : Set α → E →L[ℝ] F) 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} (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S 0 f = 0
-- MeasureTheory.L1.SimpleFunc.setToL1S_zero_left'
theorem setToL1S_zero_left' {T : Set α → E →L[ℝ] F}
    (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁ₛ[μ] E) : setToL1S T 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} {T : Set α → E →L[ℝ] F} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f = 0
-- MeasureTheory.L1.SimpleFunc.setToL1S_congr
theorem setToL1S_congr (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T) {f g : α →₁ₛ[μ] E} (h : toSimpleFunc f =ᵐ[μ] toSimpleFunc g) :
    setToL1S T f = setToL1S T g 
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) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) {f g : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)} (h : ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) =ᵐ[μ] ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc g))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f = MeasureTheory.L1.SimpleFunc.setToL1S T g
-- MeasureTheory.L1.SimpleFunc.setToL1S_congr_left
theorem setToL1S_congr_left (T T' : Set α → E →L[ℝ] F)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁ₛ[μ] E) :
    setToL1S T f = setToL1S T' 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) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f = MeasureTheory.L1.SimpleFunc.setToL1S T' f
-- MeasureTheory.L1.SimpleFunc.setToL1S_congr_measure
theorem setToL1S_congr_measure {μ' : Measure α} (T : Set α → E →L[ℝ] F)
    (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (hμ : μ ≪ μ')
    (f : α →₁ₛ[μ] E) (f' : α →₁ₛ[μ'] E) (h : (f : α → E) =ᵐ[μ] f') :
    setToL1S T f = setToL1S T 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 μ' : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (hμ : μ ≪ μ') (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)) (f' : ↥(MeasureTheory.Lp.simpleFunc E 1 μ')) (h : ↑↑↑f =ᵐ[μ] ↑↑↑f')
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f = MeasureTheory.L1.SimpleFunc.setToL1S T f'
-- MeasureTheory.L1.SimpleFunc.setToL1S_add_left
theorem setToL1S_add_left (T T' : Set α → E →L[ℝ] F) (f : α →₁ₛ[μ] E) :
    setToL1S (T + T') f = setToL1S T f + setToL1S T' 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) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S (T + T') f =
  MeasureTheory.L1.SimpleFunc.setToL1S T f + MeasureTheory.L1.SimpleFunc.setToL1S T' f
-- MeasureTheory.L1.SimpleFunc.setToL1S_add_left'
theorem setToL1S_add_left' (T T' T'' : Set α → E →L[ℝ] F)
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁ₛ[μ] E) :
    setToL1S T'' f = setToL1S T f + setToL1S T' 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) (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.setToL1S T'' f =
  MeasureTheory.L1.SimpleFunc.setToL1S T f + MeasureTheory.L1.SimpleFunc.setToL1S T' f
-- MeasureTheory.L1.SimpleFunc.setToL1S_smul_left
theorem setToL1S_smul_left (T : Set α → E →L[ℝ] F) (c : ℝ) (f : α →₁ₛ[μ] E) :
    setToL1S (fun s => c • T s) f = c • setToL1S T 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) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S (fun s => c • T s) f = c • MeasureTheory.L1.SimpleFunc.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.setToL1S_smul_left'
theorem setToL1S_smul_left' (T T' : Set α → E →L[ℝ] F) (c : ℝ)
    (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁ₛ[μ] E) :
    setToL1S T' f = c • setToL1S T 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 : ℝ) (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.setToL1S T' f = c • MeasureTheory.L1.SimpleFunc.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.setToL1S_add
theorem setToL1S_add (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T) (f g : α →₁ₛ[μ] E) :
    setToL1S T (f + g) = setToL1S T f + setToL1S T g 
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) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (f g : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (f + g) =
  MeasureTheory.L1.SimpleFunc.setToL1S T f + MeasureTheory.L1.SimpleFunc.setToL1S T g
-- MeasureTheory.L1.SimpleFunc.setToL1S_neg
theorem setToL1S_neg {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T) (f : α →₁ₛ[μ] E) : setToL1S T (-f) = -setToL1S T 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} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (-f) = -MeasureTheory.L1.SimpleFunc.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.setToL1S_sub
theorem setToL1S_sub {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T) (f g : α →₁ₛ[μ] E) :
    setToL1S T (f - g) = setToL1S T f - setToL1S T g 
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} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (f g : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (f - g) =
  MeasureTheory.L1.SimpleFunc.setToL1S T f - MeasureTheory.L1.SimpleFunc.setToL1S T g
-- MeasureTheory.L1.SimpleFunc.setToL1S_smul_real
theorem setToL1S_smul_real (T : Set α → E →L[ℝ] F)
    (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (c : ℝ)
    (f : α →₁ₛ[μ] E) : setToL1S T (c • f) = c • setToL1S T 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) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (c : ℝ) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (c • f) = c • MeasureTheory.L1.SimpleFunc.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.setToL1S_smul
theorem setToL1S_smul {E} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E]
    [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α →₁ₛ[μ] E) : setToL1S T (c • f) = c • setToL1S T f 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedField 𝕜] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) (c : 𝕜) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (c • f) = c • MeasureTheory.L1.SimpleFunc.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.norm_setToL1S_le
theorem norm_setToL1S_le (T : Set α → E →L[ℝ] F) {C : ℝ}
    (hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) (f : α →₁ₛ[μ] E) :
    ‖setToL1S T 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} (T : Set α → E →L[ℝ] F) {C : ℝ} (hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: ‖MeasureTheory.L1.SimpleFunc.setToL1S T f‖ ≤ C * ‖f‖
-- MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst
theorem setToL1S_indicatorConst {T : Set α → E →L[ℝ] F} {s : Set α}
    (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T)
    (hs : MeasurableSet s) (hμs : μ s < ∞) (x : E) :
    setToL1S T (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} {T : Set α → E →L[ℝ] F} {s : Set α} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (hs : MeasurableSet s) (hμs : μ s < ⊤) (x : E)
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (MeasureTheory.Lp.simpleFunc.indicatorConst 1 hs ⋯ x) = (T s) x
-- MeasureTheory.L1.SimpleFunc.setToL1S_const
theorem setToL1S_const [IsFiniteMeasure μ] {T : Set α → E →L[ℝ] F}
    (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) (x : E) :
    setToL1S T (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} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (x : E)
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T (MeasureTheory.Lp.simpleFunc.indicatorConst 1 ⋯ ⋯ x) = (T Set.univ) x
-- MeasureTheory.L1.SimpleFunc.setToL1S_mono_left
theorem setToL1S_mono_left {T T' : Set α → E →L[ℝ] G''} (hTT' : ∀ s x, T s x ≤ T' s x)
    (f : α →₁ₛ[μ] E) : setToL1S T f ≤ setToL1S T' 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_7} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {T T' : Set α → E →L[ℝ] G''} (hTT' : ∀ (s : Set α) (x : E), (T s) x ≤ (T' s) x) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.setToL1S T f ≤ MeasureTheory.L1.SimpleFunc.setToL1S T' f
-- MeasureTheory.L1.SimpleFunc.setToL1S_mono_left'
theorem setToL1S_mono_left' {T T' : Set α → E →L[ℝ] G''}
    (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →₁ₛ[μ] E) :
    setToL1S T f ≤ setToL1S T' 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_7} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {T T' : Set α → E →L[ℝ] G''} (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.setToL1S T f ≤ MeasureTheory.L1.SimpleFunc.setToL1S T' f
-- MeasureTheory.L1.SimpleFunc.setToL1S_nonneg
theorem setToL1S_nonneg (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f : α →₁ₛ[μ] G''}
    (hf : 0 ≤ f) : 0 ≤ setToL1S T 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'} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (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.setToL1S T f
-- MeasureTheory.L1.SimpleFunc.setToL1S_mono
theorem setToL1S_mono (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
    (h_add : FinMeasAdditive μ T)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →₁ₛ[μ] G''}
    (hfg : f ≤ g) : setToL1S T f ≤ setToL1S T 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'} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) (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.setToL1S T f ≤ MeasureTheory.L1.SimpleFunc.setToL1S T g
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left
@[simp]
theorem setToL1SCLM_zero_left (hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C)
    (f : α →₁ₛ[μ] E) : setToL1SCLM α E μ 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} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ 0 C) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f = 0
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left'
theorem setToL1SCLM_zero_left' (hT : DominatedFinMeasAdditive μ T C)
    (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ 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} {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.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f = 0
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left
theorem setToL1SCLM_congr_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (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} {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} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (h : T = T') (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_congr_left'
theorem setToL1SCLM_congr_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s)
    (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} {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} (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.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_congr_measure
theorem setToL1SCLM_congr_measure {μ' : Measure α} (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ' T C') (hμ : μ ≪ μ') (f : α →₁ₛ[μ] E) (f' : α →₁ₛ[μ'] E)
    (h : (f : α → E) =ᵐ[μ] 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 : Set α → E →L[ℝ] F} {C C' : ℝ} {μ' : MeasureTheory.Measure α} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ' T C') (hμ : μ ≪ μ') (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)) (f' : ↥(MeasureTheory.Lp.simpleFunc E 1 μ')) (h : ↑↑↑f =ᵐ[μ] ↑↑↑f')
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f = (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ' hT') f'
-- MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left
theorem setToL1SCLM_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁ₛ[μ] E) :
    setToL1SCLM α E μ (hT.add 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' : Set α → E →L[ℝ] F} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C') (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ ⋯) f =
  (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f + (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f