ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.continuous_L1_toL1
theorem continuous_L1_toL1 {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞) (hμ'_le : μ' ≤ c' • μ) :
    Continuous fun f : α →₁[μ] G =>
      (Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f)).toL1 f 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m : MeasurableSpace α} {μ μ' : MeasureTheory.Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ⊤) (hμ'_le : μ' ≤ c' • μ)
Goal: Continuous fun f => MeasureTheory.Integrable.toL1 ↑↑f ⋯
-- MeasureTheory.setToFun_congr_measure_of_integrable
theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞)
    (hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
    (hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) :
    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 C' : ℝ} {μ' : MeasureTheory.Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ⊤) (hμ'_le : μ' ≤ c' • μ) (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun μ' T hT' f
-- MeasureTheory.setToFun_congr_measure
theorem setToFun_congr_measure {μ' : Measure α} (c c' : ℝ≥0∞) (hc : c ≠ ∞) (hc' : c' ≠ ∞)
    (hμ_le : μ ≤ c • μ') (hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
    (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 C' : ℝ} {μ' : MeasureTheory.Measure α} (c c' : ℝ≥0∞) (hc : c ≠ ⊤) (hc' : c' ≠ ⊤) (hμ_le : μ ≤ c • μ') (hμ'_le : μ' ≤ c' • μ) (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ' T C') (f : α → E)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun μ' T hT' f
-- MeasureTheory.setToFun_congr_measure_of_add_right
theorem setToFun_congr_measure_of_add_right {μ' : Measure α}
    (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ T C)
    (f : α → E) (hf : Integrable f (μ + μ')) :
    setToFun (μ + μ') T hT_add 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 C' : ℝ} {μ' : MeasureTheory.Measure α} (hT_add : MeasureTheory.DominatedFinMeasAdditive (μ + μ') T C') (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (f : α → E) (hf : MeasureTheory.Integrable f (μ + μ'))
Goal: MeasureTheory.setToFun (μ + μ') T hT_add f = MeasureTheory.setToFun μ T hT f
-- MeasureTheory.setToFun_congr_measure_of_add_left
theorem setToFun_congr_measure_of_add_left {μ' : Measure α}
    (hT_add : DominatedFinMeasAdditive (μ + μ') T C') (hT : DominatedFinMeasAdditive μ' T C)
    (f : α → E) (hf : Integrable f (μ + μ')) :
    setToFun (μ + μ') T hT_add 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 C' : ℝ} {μ' : MeasureTheory.Measure α} (hT_add : MeasureTheory.DominatedFinMeasAdditive (μ + μ') T C') (hT : MeasureTheory.DominatedFinMeasAdditive μ' T C) (f : α → E) (hf : MeasureTheory.Integrable f (μ + μ'))
Goal: MeasureTheory.setToFun (μ + μ') T hT_add f = MeasureTheory.setToFun μ' T hT f
-- MeasureTheory.setToFun_top_smul_measure
theorem setToFun_top_smul_measure (hT : DominatedFinMeasAdditive (∞ • μ) T C) (f : α → E) :
    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} (hT : MeasureTheory.DominatedFinMeasAdditive (⊤ • μ) T C) (f : α → E)
Goal: MeasureTheory.setToFun (⊤ • μ) T hT f = 0
-- MeasureTheory.setToFun_congr_smul_measure
theorem setToFun_congr_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞)
    (hT : DominatedFinMeasAdditive μ T C) (hT_smul : DominatedFinMeasAdditive (c • μ) T C')
    (f : α → E) : setToFun μ T hT f = setToFun (c • μ) T hT_smul 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 C' : ℝ} (c : ℝ≥0∞) (hc_ne_top : c ≠ ⊤) (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT_smul : MeasureTheory.DominatedFinMeasAdditive (c • μ) T C') (f : α → E)
Goal: MeasureTheory.setToFun μ T hT f = MeasureTheory.setToFun (c • μ) T hT_smul f
-- MeasureTheory.norm_setToFun_le_mul_norm
theorem norm_setToFun_le_mul_norm (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E)
    (hC : 0 ≤ C) : ‖setToFun μ T 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) (f : ↥(MeasureTheory.Lp E 1 μ)) (hC : 0 ≤ C)
Goal: ‖MeasureTheory.setToFun μ T hT ↑↑f‖ ≤ C * ‖f‖
-- MeasureTheory.norm_setToFun_le_mul_norm'
theorem norm_setToFun_le_mul_norm' (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) :
    ‖setToFun μ T 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.setToFun μ T hT ↑↑f‖ ≤ max C 0 * ‖f‖
-- MeasureTheory.norm_setToFun_le
theorem norm_setToFun_le (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hC : 0 ≤ C) :
    ‖setToFun μ T hT f‖ ≤ C * ‖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) (hC : 0 ≤ C)
Goal: ‖MeasureTheory.setToFun μ T hT f‖ ≤ C * ‖MeasureTheory.Integrable.toL1 f hf‖
-- MeasureTheory.norm_setToFun_le'
theorem norm_setToFun_le' (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) :
    ‖setToFun μ T hT f‖ ≤ max C 0 * ‖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‖ ≤ max C 0 * ‖MeasureTheory.Integrable.toL1 f hf‖
-- MeasureTheory.tendsto_setToFun_of_dominated_convergence
theorem tendsto_setToFun_of_dominated_convergence (hT : DominatedFinMeasAdditive μ T C)
    {fs : ℕ → α → E} {f : α → E} (bound : α → ℝ)
    (fs_measurable : ∀ n, AEStronglyMeasurable (fs n) μ) (bound_integrable : Integrable bound μ)
    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => setToFun μ T hT (fs 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) {fs : ℕ → α → E} {f : α → E} (bound : α → ℝ) (fs_measurable : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (fs n) μ) (bound_integrable : MeasureTheory.Integrable bound μ) (h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => fs n a) Filter.atTop (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT (fs n)) Filter.atTop (𝓝 (MeasureTheory.setToFun μ T hT f))
-- MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence
theorem tendsto_setToFun_filter_of_dominated_convergence (hT : DominatedFinMeasAdditive μ T C) {ι}
    {l : Filter ι} [l.IsCountablyGenerated] {fs : ι → α → E} {f : α → E} (bound : α → ℝ)
    (hfs_meas : ∀ᶠ n in l, AEStronglyMeasurable (fs n) μ)
    (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => fs n a) l (𝓝 (f a))) :
    Tendsto (fun n => setToFun μ T hT (fs n)) 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} {l : Filter ι} [l.IsCountablyGenerated] {fs : ι → α → E} {f : α → E} (bound : α → ℝ) (hfs_meas : ∀ᶠ (n : ι) in l, MeasureTheory.AEStronglyMeasurable (fs n) μ) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => fs n a) l (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT (fs n)) l (𝓝 (MeasureTheory.setToFun μ T hT f))
-- MeasureTheory.continuousWithinAt_setToFun_of_dominated
theorem continuousWithinAt_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C)
    {fs : X → α → E} {x₀ : X} {bound : α → ℝ} {s : Set X}
    (hfs_meas : ∀ᶠ x in 𝓝[s] x₀, AEStronglyMeasurable (fs x) μ)
    (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ a ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousWithinAt (fun x => fs x a) s x₀) :
    ContinuousWithinAt (fun x => setToFun μ T hT (fs x)) s x₀ 
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} {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {fs : X → α → E} {x₀ : X} {bound : α → ℝ} {s : Set X} (hfs_meas : ∀ᶠ (x : X) in 𝓝[s] x₀, MeasureTheory.AEStronglyMeasurable (fs x) μ) (h_bound : ∀ᶠ (x : X) in 𝓝[s] x₀, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => fs x a) s x₀)
Goal: ContinuousWithinAt (fun x => MeasureTheory.setToFun μ T hT (fs x)) s x₀
-- MeasureTheory.continuousAt_setToFun_of_dominated
theorem continuousAt_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C) {fs : X → α → E}
    {x₀ : X} {bound : α → ℝ} (hfs_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (fs x) μ)
    (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ a ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousAt (fun x => fs x a) x₀) :
    ContinuousAt (fun x => setToFun μ T hT (fs x)) x₀ 
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} {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {fs : X → α → E} {x₀ : X} {bound : α → ℝ} (hfs_meas : ∀ᶠ (x : X) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (fs x) μ) (h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousAt (fun x => fs x a) x₀)
Goal: ContinuousAt (fun x => MeasureTheory.setToFun μ T hT (fs x)) x₀
-- MeasureTheory.continuousOn_setToFun_of_dominated
theorem continuousOn_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C) {fs : X → α → E}
    {bound : α → ℝ} {s : Set X} (hfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ)
    (h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, ContinuousOn (fun x => fs x a) s) :
    ContinuousOn (fun x => setToFun μ T hT (fs x)) s 
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} {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {fs : X → α → E} {bound : α → ℝ} {s : Set X} (hfs_meas : ∀ x ∈ s, MeasureTheory.AEStronglyMeasurable (fs x) μ) (h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s)
Goal: ContinuousOn (fun x => MeasureTheory.setToFun μ T hT (fs x)) s
-- MeasureTheory.continuous_setToFun_of_dominated
theorem continuous_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C) {fs : X → α → E}
    {bound : α → ℝ} (hfs_meas : ∀ x, AEStronglyMeasurable (fs x) μ)
    (h_bound : ∀ x, ∀ᵐ a ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
    (h_cont : ∀ᵐ a ∂μ, Continuous fun x => fs x a) : Continuous fun x => setToFun μ T hT (fs 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} {X : Type u_7} [TopologicalSpace X] [FirstCountableTopology X] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {fs : X → α → E} {bound : α → ℝ} (hfs_meas : ∀ (x : X), MeasureTheory.AEStronglyMeasurable (fs x) μ) (h_bound : ∀ (x : X), ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : MeasureTheory.Integrable bound μ) (h_cont : ∀ᵐ (a : α) ∂μ, Continuous fun x => fs x a)
Goal: Continuous fun x => MeasureTheory.setToFun μ T hT (fs x)
-- MeasureTheory.Measure.restrictₗ_apply
@[simp]
theorem restrictₗ_apply {_m0 : MeasurableSpace α} (s : Set α) (μ : Measure α) :
    restrictₗ s μ = μ.restrict s 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} (s : Set α) (μ : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.restrictₗ s) μ = μ.restrict s
-- MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict
theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
    (μ.restrict s).toOuterMeasure = OuterMeasure.restrict s μ.toOuterMeasure 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasurableSet s)
Goal: (μ.restrict s).toOuterMeasure = (MeasureTheory.OuterMeasure.restrict s) μ.toOuterMeasure
-- MeasureTheory.Measure.restrict_apply₀
theorem restrict_apply₀ (ht : NullMeasurableSet t (μ.restrict s)) : μ.restrict s t = μ (t ∩ s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasureTheory.NullMeasurableSet t (μ.restrict s))
Goal: (μ.restrict s) t = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_apply
@[simp]
theorem restrict_apply (ht : MeasurableSet t) : μ.restrict s t = μ (t ∩ s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict s) t = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_mono'
theorem restrict_mono' {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ ⦃μ ν : Measure α⦄ (hs : s ≤ᵐ[μ] s')
    (hμν : μ ≤ ν) : μ.restrict s ≤ ν.restrict s' 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ ⦃μ ν : MeasureTheory.Measure α⦄ (hs : s ≤ᵐ[μ] s') (hμν : μ ≤ ν)
Goal: μ.restrict s ≤ ν.restrict s'
-- MeasureTheory.Measure.restrict_mono
@[mono]
theorem restrict_mono {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ (hs : s ⊆ s') ⦃μ ν : Measure α⦄
    (hμν : μ ≤ ν) : μ.restrict s ≤ ν.restrict s' 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ (hs : s ⊆ s') ⦃μ ν : MeasureTheory.Measure α⦄ (hμν : μ ≤ ν)
Goal: μ.restrict s ≤ ν.restrict s'
-- MeasureTheory.Measure.restrict_mono_ae
theorem restrict_mono_ae (h : s ≤ᵐ[μ] t) : μ.restrict s ≤ μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : s ≤ᵐ[μ] t)
Goal: μ.restrict s ≤ μ.restrict t
-- MeasureTheory.Measure.restrict_congr_set
theorem restrict_congr_set (h : s =ᵐ[μ] t) : μ.restrict s = μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : s =ᵐ[μ] t)
Goal: μ.restrict s = μ.restrict t
-- MeasureTheory.Measure.restrict_apply'
@[simp]
theorem restrict_apply' (hs : MeasurableSet s) : μ.restrict s t = μ (t ∩ s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict s) t = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_apply₀'
theorem restrict_apply₀' (hs : NullMeasurableSet s μ) : μ.restrict s t = μ (t ∩ s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: (μ.restrict s) t = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_le_self
theorem restrict_le_self : μ.restrict s ≤ μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: μ.restrict s ≤ μ
-- MeasureTheory.Measure.restrict_eq_self
theorem restrict_eq_self (h : s ⊆ t) : μ.restrict t s = μ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s t : Set α} (h : s ⊆ t)
Goal: (μ.restrict t) s = μ s
-- MeasureTheory.Measure.restrict_apply_self
@[simp]
theorem restrict_apply_self (s : Set α) : (μ.restrict s) s = μ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α)
Goal: (μ.restrict s) s = μ s
-- MeasureTheory.Measure.restrict_apply_univ
theorem restrict_apply_univ (s : Set α) : μ.restrict s univ = μ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α)
Goal: (μ.restrict s) Set.univ = μ s
-- MeasureTheory.Measure.le_restrict_apply
theorem le_restrict_apply (s t : Set α) : μ (t ∩ s) ≤ μ.restrict s t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α)
Goal: μ (t ∩ s) ≤ (μ.restrict s) t
-- MeasureTheory.Measure.restrict_apply_le
theorem restrict_apply_le (s t : Set α) : μ.restrict s t ≤ μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α)
Goal: (μ.restrict s) t ≤ μ t
-- MeasureTheory.Measure.restrict_apply_superset
theorem restrict_apply_superset (h : s ⊆ t) : μ.restrict s t = μ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : s ⊆ t)
Goal: (μ.restrict s) t = μ s
-- MeasureTheory.Measure.restrict_add
@[simp]
theorem restrict_add {_m0 : MeasurableSpace α} (μ ν : Measure α) (s : Set α) :
    (μ + ν).restrict s = μ.restrict s + ν.restrict s 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) (s : Set α)
Goal: (μ + ν).restrict s = μ.restrict s + ν.restrict s
-- MeasureTheory.Measure.restrict_zero
@[simp]
theorem restrict_zero {_m0 : MeasurableSpace α} (s : Set α) : (0 : Measure α).restrict s = 0 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} (s : Set α)
Goal: MeasureTheory.Measure.restrict 0 s = 0
-- MeasureTheory.Measure.restrict_smul
@[simp]
theorem restrict_smul {_m0 : MeasurableSpace α} (c : ℝ≥0∞) (μ : Measure α) (s : Set α) :
    (c • μ).restrict s = c • μ.restrict s 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} (c : ℝ≥0∞) (μ : MeasureTheory.Measure α) (s : Set α)
Goal: (c • μ).restrict s = c • μ.restrict s
-- MeasureTheory.Measure.restrict_restrict₀
theorem restrict_restrict₀ (hs : NullMeasurableSet s (μ.restrict t)) :
    (μ.restrict t).restrict s = μ.restrict (s ∩ t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s (μ.restrict t))
Goal: (μ.restrict t).restrict s = μ.restrict (s ∩ t)
-- MeasureTheory.Measure.restrict_restrict
@[simp]
theorem restrict_restrict (hs : MeasurableSet s) : (μ.restrict t).restrict s = μ.restrict (s ∩ t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict t).restrict s = μ.restrict (s ∩ t)
-- MeasureTheory.Measure.restrict_restrict_of_subset
theorem restrict_restrict_of_subset (h : s ⊆ t) : (μ.restrict t).restrict s = μ.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : s ⊆ t)
Goal: (μ.restrict t).restrict s = μ.restrict s
-- MeasureTheory.Measure.restrict_restrict₀'
theorem restrict_restrict₀' (ht : NullMeasurableSet t μ) :
    (μ.restrict t).restrict s = μ.restrict (s ∩ t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: (μ.restrict t).restrict s = μ.restrict (s ∩ t)
-- MeasureTheory.Measure.restrict_restrict'
theorem restrict_restrict' (ht : MeasurableSet t) :
    (μ.restrict t).restrict s = μ.restrict (s ∩ t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict t).restrict s = μ.restrict (s ∩ t)
-- MeasureTheory.Measure.restrict_comm
theorem restrict_comm (hs : MeasurableSet s) :
    (μ.restrict t).restrict s = (μ.restrict s).restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict t).restrict s = (μ.restrict s).restrict t
-- MeasureTheory.Measure.restrict_apply_eq_zero
theorem restrict_apply_eq_zero (ht : MeasurableSet t) : μ.restrict s t = 0 ↔ μ (t ∩ s) = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict s) t = 0 ↔ μ (t ∩ s) = 0
-- MeasureTheory.Measure.measure_inter_eq_zero_of_restrict
theorem measure_inter_eq_zero_of_restrict (h : μ.restrict s t = 0) : μ (t ∩ s) = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : (μ.restrict s) t = 0)
Goal: μ (t ∩ s) = 0
-- MeasureTheory.Measure.restrict_apply_eq_zero'
theorem restrict_apply_eq_zero' (hs : MeasurableSet s) : μ.restrict s t = 0 ↔ μ (t ∩ s) = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s)
Goal: (μ.restrict s) t = 0 ↔ μ (t ∩ s) = 0
-- MeasureTheory.Measure.restrict_eq_zero
@[simp]
theorem restrict_eq_zero : μ.restrict s = 0 ↔ μ s = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: μ.restrict s = 0 ↔ μ s = 0
-- MeasureTheory.Measure.restrict_zero_set
theorem restrict_zero_set {s : Set α} (h : μ s = 0) : μ.restrict s = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : μ s = 0)
Goal: μ.restrict s = 0
-- MeasureTheory.Measure.restrict_empty
@[simp]
theorem restrict_empty : μ.restrict ∅ = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.restrict ∅ = 0
-- MeasureTheory.Measure.restrict_univ
@[simp]
theorem restrict_univ : μ.restrict univ = μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.restrict Set.univ = μ