ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.L1.tendsto_setToL1
theorem tendsto_setToL1 (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) {ι}
    (fs : ι → α →₁[μ] E) {l : Filter ι} (hfs : Tendsto fs l (𝓝 f)) :
    Tendsto (fun i => setToL1 hT (fs i)) l (𝓝 <| 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) (f : ↥(MeasureTheory.Lp E 1 μ)) {ι : Type u_7} (fs : ι → ↥(MeasureTheory.Lp E 1 μ)) {l : Filter ι} (hfs : Filter.Tendsto fs l (𝓝 f))
Goal: Filter.Tendsto (fun i => (MeasureTheory.L1.setToL1 hT) (fs i)) l (𝓝 ((MeasureTheory.L1.setToL1 hT) f))
-- MeasureTheory.setToFun_eq
theorem setToFun_eq (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
    setToFun μ T hT f = L1.setToL1 hT (hf.toL1 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.setToFun μ T hT f = (MeasureTheory.L1.setToL1 hT) (MeasureTheory.Integrable.toL1 f hf)
-- MeasureTheory.L1.setToFun_eq_setToL1
theorem L1.setToFun_eq_setToL1 (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) :
    setToFun μ T hT f = L1.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) (f : ↥(MeasureTheory.Lp E 1 μ))
Goal: MeasureTheory.setToFun μ T hT ↑↑f = (MeasureTheory.L1.setToL1 hT) f
-- MeasureTheory.setToFun_undef
theorem setToFun_undef (hT : DominatedFinMeasAdditive μ T C) (hf : ¬Integrable f μ) :
    setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : ¬MeasureTheory.Integrable f μ)
Goal: MeasureTheory.setToFun μ T hT f = 0
-- MeasureTheory.setToFun_non_aEStronglyMeasurable
theorem setToFun_non_aEStronglyMeasurable (hT : DominatedFinMeasAdditive μ T C)
    (hf : ¬AEStronglyMeasurable f μ) : setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : ¬MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.setToFun μ T hT f = 0
-- MeasureTheory.setToFun_congr_left
theorem setToFun_congr_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : α → E) :
    setToFun μ T hT f = setToFun μ T' 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 : α → E)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun μ T' hT' f
-- MeasureTheory.setToFun_congr_left'
theorem setToFun_congr_left' (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s)
    (f : α → E) : setToFun μ T hT f = setToFun μ T' 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 : α → E)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun μ T' hT' f
-- MeasureTheory.setToFun_add_left
theorem setToFun_add_left (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ T' C') (f : α → E) :
    setToFun μ (T + T') (hT.add hT') f = setToFun μ T hT f + setToFun μ T' 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 : α → E)
Goal: MeasureTheory.setToFun μ (T + T') ⋯ f = MeasureTheory.setToFun μ T hT f + MeasureTheory.setToFun μ T' hT' f
-- MeasureTheory.setToFun_add_left'
theorem setToFun_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 → E) :
    setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' 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 : α → E)
Goal: MeasureTheory.setToFun μ T'' hT'' f = MeasureTheory.setToFun μ T hT f + MeasureTheory.setToFun μ T' hT' f
-- MeasureTheory.setToFun_smul_left
theorem setToFun_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α → E) :
    setToFun μ (fun s => c • T s) (hT.smul c) f = c • setToFun μ T 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 : α → E)
Goal: MeasureTheory.setToFun μ (fun s => c • T s) ⋯ f = c • MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_smul_left'
theorem setToFun_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 → E) :
    setToFun μ T' hT' f = c • setToFun μ T 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 : α → E)
Goal: MeasureTheory.setToFun μ T' hT' f = c • MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_zero
@[simp]
theorem setToFun_zero (hT : DominatedFinMeasAdditive μ T C) : setToFun μ T hT (0 : α → E) = 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.setToFun μ T hT 0 = 0
-- MeasureTheory.setToFun_zero_left
@[simp]
theorem setToFun_zero_left {hT : DominatedFinMeasAdditive μ (0 : Set α → E →L[ℝ] F) C} :
    setToFun μ 0 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} {f : α → E} {hT : MeasureTheory.DominatedFinMeasAdditive μ 0 C}
Goal: MeasureTheory.setToFun μ 0 hT f = 0
-- MeasureTheory.setToFun_zero_left'
theorem setToFun_zero_left' (hT : DominatedFinMeasAdditive μ T C)
    (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) : setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0)
Goal: MeasureTheory.setToFun μ T hT f = 0
-- MeasureTheory.setToFun_add
theorem setToFun_add (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ)
    (hg : Integrable g μ) : setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT 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} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} {f g : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.setToFun μ T hT (f + g) = MeasureTheory.setToFun μ T hT f + MeasureTheory.setToFun μ T hT g
-- MeasureTheory.setToFun_finset_sum'
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, setToFun μ T hT (f i) 
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) {ι : Type u_7} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: MeasureTheory.setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, MeasureTheory.setToFun μ T hT (f i)
-- MeasureTheory.setToFun_finset_sum
theorem setToFun_finset_sum (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι) {f : ι → α → E}
    (hf : ∀ i ∈ s, Integrable (f i) μ) :
    (setToFun μ T hT fun a => ∑ i ∈ s, f i a) = ∑ i ∈ s, setToFun μ T hT (f i) 
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) {ι : Type u_7} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: (MeasureTheory.setToFun μ T hT fun a => ∑ i ∈ s, f i a) = ∑ i ∈ s, MeasureTheory.setToFun μ T hT (f i)
-- MeasureTheory.setToFun_neg
theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
    setToFun μ T hT (-f) = -setToFun μ T 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 : α → E)
Goal: MeasureTheory.setToFun μ T hT (-f) = -MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_sub
theorem setToFun_sub (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ)
    (hg : Integrable g μ) : setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT 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} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} {f g : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.setToFun μ T hT (f - g) = MeasureTheory.setToFun μ T hT f - MeasureTheory.setToFun μ T hT g
-- MeasureTheory.setToFun_smul
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T 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} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (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 : α → E)
Goal: MeasureTheory.setToFun μ T hT (c • f) = c • MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_congr_ae
theorem setToFun_congr_ae (hT : DominatedFinMeasAdditive μ T C) (h : f =ᵐ[μ] g) :
    setToFun μ T hT f = setToFun μ T hT 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} [CompleteSpace F] {T : Set α → E →L[ℝ] F} {C : ℝ} {f g : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h : f =ᵐ[μ] g)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun μ T hT g
-- MeasureTheory.setToFun_measure_zero
theorem setToFun_measure_zero (hT : DominatedFinMeasAdditive μ T C) (h : μ = 0) :
    setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h : μ = 0)
Goal: MeasureTheory.setToFun μ T hT f = 0
-- MeasureTheory.setToFun_measure_zero'
theorem setToFun_measure_zero' (hT : DominatedFinMeasAdditive μ T C)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → μ s = 0) : setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → μ s = 0)
Goal: MeasureTheory.setToFun μ T hT f = 0
-- MeasureTheory.setToFun_toL1
theorem setToFun_toL1 (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
    setToFun μ T hT (hf.toL1 f) = setToFun μ T 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} {f : α → E} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.setToFun μ T hT ↑↑(MeasureTheory.Integrable.toL1 f hf) = MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_indicator_const
theorem setToFun_indicator_const (hT : DominatedFinMeasAdditive μ T C) {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) :
    setToFun μ T hT (s.indicator fun _ => 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.setToFun μ T hT (s.indicator fun x_1 => x) = (T s) x
-- MeasureTheory.setToFun_const
theorem setToFun_const [IsFiniteMeasure μ] (hT : DominatedFinMeasAdditive μ T C) (x : E) :
    (setToFun μ T hT fun _ => 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.setToFun μ T hT fun x_1 => x) = (T Set.univ) x
-- MeasureTheory.setToFun_mono_left'
theorem setToFun_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 → E) :
    setToFun μ T hT f ≤ setToFun μ T' 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 : α → E)
Goal: MeasureTheory.setToFun μ T hT f ≤ MeasureTheory.setToFun μ T' hT' f
-- MeasureTheory.setToFun_mono_left
theorem setToFun_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) : setToFun μ T hT f ≤ setToFun μ T' 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.setToFun μ T hT ↑↑f ≤ MeasureTheory.setToFun μ T' hT' ↑↑f
-- MeasureTheory.setToFun_nonneg
theorem setToFun_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 → G'}
    (hf : 0 ≤ᵐ[μ] f) : 0 ≤ setToFun μ T 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 : α → G'} (hf : 0 ≤ᵐ[μ] f)
Goal: 0 ≤ MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_mono
theorem setToFun_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 → G'}
    (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    setToFun μ T hT f ≤ setToFun μ T 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 : α → G'} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : f ≤ᵐ[μ] g)
Goal: MeasureTheory.setToFun μ T hT f ≤ MeasureTheory.setToFun μ T hT g
-- MeasureTheory.continuous_setToFun
@[continuity]
theorem continuous_setToFun (hT : DominatedFinMeasAdditive μ T C) :
    Continuous fun f : α →₁[μ] E => setToFun μ T 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)
Goal: Continuous fun f => MeasureTheory.setToFun μ T hT ↑↑f
-- MeasureTheory.tendsto_setToFun_of_L1
theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
    (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
    Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T 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) {ι : Type u_7} (f : α → E) (hfi : MeasureTheory.Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ (i : ι) in l, MeasureTheory.Integrable (fs i) μ) (hfs : Filter.Tendsto (fun i => ∫⁻ (x : α), ↑‖fs i x - f x‖₊ ∂μ) l (𝓝 0))
Goal: Filter.Tendsto (fun i => MeasureTheory.setToFun μ T hT (fs i)) l (𝓝 (MeasureTheory.setToFun μ T hT f))
-- MeasureTheory.tendsto_setToFun_approxOn_of_measurable
theorem tendsto_setToFun_approxOn_of_measurable (hT : DominatedFinMeasAdditive μ T C)
    [MeasurableSpace E] [BorelSpace E] {f : α → E} {s : Set E} [SeparableSpace s]
    (hfi : Integrable f μ) (hfm : Measurable f) (hs : ∀ᵐ x ∂μ, f x ∈ closure s) {y₀ : E}
    (h₀ : y₀ ∈ s) (h₀i : Integrable (fun _ => y₀) μ) :
    Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f hfm s y₀ h₀ n)) atTop
      (𝓝 <| setToFun μ T 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) [MeasurableSpace E] [BorelSpace E] {f : α → E} {s : Set E} [TopologicalSpace.SeparableSpace ↑s] (hfi : MeasureTheory.Integrable f μ) (hfm : Measurable f) (hs : ∀ᵐ (x : α) ∂μ, f x ∈ closure s) {y₀ : E} (h₀ : y₀ ∈ s) (h₀i : MeasureTheory.Integrable (fun x => y₀) μ)
Goal: Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT ↑(MeasureTheory.SimpleFunc.approxOn f hfm s y₀ h₀ n))
  Filter.atTop (𝓝 (MeasureTheory.setToFun μ T hT f))
-- MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset
theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
    (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
    (fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
    (hs : range f ∪ {0} ⊆ s) :
    Tendsto (fun n => setToFun μ T hT (SimpleFunc.approxOn f fmeas s 0 (hs <| by simp) n)) atTop
      (𝓝 <| setToFun μ T 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) [MeasurableSpace E] [BorelSpace E] {f : α → E} (fmeas : Measurable f) (hf : MeasureTheory.Integrable f μ) (s : Set E) [TopologicalSpace.SeparableSpace ↑s] (hs : Set.range f ∪ {0} ⊆ s)
Goal: Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT ↑(MeasureTheory.SimpleFunc.approxOn f fmeas s 0 ⋯ n))
  Filter.atTop (𝓝 (MeasureTheory.setToFun μ T hT f))