ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integral_add_compl₀
theorem integral_add_compl₀ (hs : NullMeasurableSet s μ) (hfi : Integrable f μ) :
    ∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasureTheory.NullMeasurableSet s μ) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ
-- MeasureTheory.integral_add_compl
theorem integral_add_compl (hs : MeasurableSet s) (hfi : Integrable f μ) :
    ∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasurableSet s) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ
-- MeasureTheory.setIntegral_compl
theorem setIntegral_compl (hs : MeasurableSet s) (hfi : Integrable f μ) :
    ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ - ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasurableSet s) (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_indicator
theorem integral_indicator (hs : MeasurableSet s) :
    ∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasurableSet s)
Goal: ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_indicator
theorem setIntegral_indicator (ht : MeasurableSet t) :
    ∫ x in s, t.indicator f x ∂μ = ∫ x in s ∩ t, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasurableSet t)
Goal: ∫ (x : X) in s, t.indicator f x ∂μ = ∫ (x : X) in s ∩ t, f x ∂μ
-- MeasureTheory.ofReal_setIntegral_one_of_measure_ne_top
theorem ofReal_setIntegral_one_of_measure_ne_top {X : Type*} {m : MeasurableSpace X}
    {μ : Measure X} {s : Set X} (hs : μ s ≠ ∞) : ENNReal.ofReal (∫ _ in s, (1 : ℝ) ∂μ) = μ s 
Arguments: {X : Type u_5} {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} {s : Set X} (hs : μ s ≠ ⊤)
Goal: ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = μ s
-- MeasureTheory.ofReal_setIntegral_one
theorem ofReal_setIntegral_one {X : Type*} {_ : MeasurableSpace X} (μ : Measure X)
    [IsFiniteMeasure μ] (s : Set X) : ENNReal.ofReal (∫ _ in s, (1 : ℝ) ∂μ) = μ s 
Arguments: {X : Type u_5}
Goal: ∀ {x : MeasurableSpace X} (μ : MeasureTheory.Measure X) [inst : MeasureTheory.IsFiniteMeasure μ] (s : Set X),
  ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = μ s
-- MeasureTheory.integral_piecewise
theorem integral_piecewise [DecidablePred (· ∈ s)] (hs : MeasurableSet s) (hf : IntegrableOn f s μ)
    (hg : IntegrableOn g sᶜ μ) :
    ∫ x, s.piecewise f g x ∂μ = ∫ x in s, f x ∂μ + ∫ x in sᶜ, g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : X → E} {s : Set X} {μ : MeasureTheory.Measure X} [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s) (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g sᶜ μ)
Goal: ∫ (x : X), s.piecewise f g x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, g x ∂μ
-- MeasureTheory.tendsto_setIntegral_of_monotone
theorem tendsto_setIntegral_of_monotone {ι : Type*} [Countable ι] [SemilatticeSup ι]
    {s : ι → Set X} (hsm : ∀ i, MeasurableSet (s i)) (h_mono : Monotone s)
    (hfi : IntegrableOn f (⋃ n, s n) μ) :
    Tendsto (fun i => ∫ x in s i, f x ∂μ) atTop (𝓝 (∫ x in ⋃ n, s n, f x ∂μ)) 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] [SemilatticeSup ι] {s : ι → Set X} (hsm : ∀ (i : ι), MeasurableSet (s i)) (h_mono : Monotone s) (hfi : MeasureTheory.IntegrableOn f (⋃ n, s n) μ)
Goal: Filter.Tendsto (fun i => ∫ (x : X) in s i, f x ∂μ) Filter.atTop (𝓝 (∫ (x : X) in ⋃ n, s n, f x ∂μ))
-- MeasureTheory.tendsto_setIntegral_of_antitone
theorem tendsto_setIntegral_of_antitone {ι : Type*} [Countable ι] [SemilatticeSup ι]
    {s : ι → Set X} (hsm : ∀ i, MeasurableSet (s i)) (h_anti : Antitone s)
    (hfi : ∃ i, IntegrableOn f (s i) μ) :
    Tendsto (fun i ↦ ∫ x in s i, f x ∂μ) atTop (𝓝 (∫ x in ⋂ n, s n, f x ∂μ)) 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] [SemilatticeSup ι] {s : ι → Set X} (hsm : ∀ (i : ι), MeasurableSet (s i)) (h_anti : Antitone s) (hfi : ∃ i, MeasureTheory.IntegrableOn f (s i) μ)
Goal: Filter.Tendsto (fun i => ∫ (x : X) in s i, f x ∂μ) Filter.atTop (𝓝 (∫ (x : X) in ⋂ n, s n, f x ∂μ))
-- MeasureTheory.hasSum_integral_iUnion_ae
theorem hasSum_integral_iUnion_ae {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) :
    HasSum (fun n => ∫ x in s n, f x ∂μ) (∫ x in ⋃ n, s n, f x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] {s : ι → Set X} (hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (hfi : MeasureTheory.IntegrableOn f (⋃ i, s i) μ)
Goal: HasSum (fun n => ∫ (x : X) in s n, f x ∂μ) (∫ (x : X) in ⋃ n, s n, f x ∂μ)
-- MeasureTheory.hasSum_integral_iUnion
theorem hasSum_integral_iUnion {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, MeasurableSet (s i)) (hd : Pairwise (Disjoint on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) :
    HasSum (fun n => ∫ x in s n, f x ∂μ) (∫ x in ⋃ n, s n, f x ∂μ) 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] {s : ι → Set X} (hm : ∀ (i : ι), MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (hfi : MeasureTheory.IntegrableOn f (⋃ i, s i) μ)
Goal: HasSum (fun n => ∫ (x : X) in s n, f x ∂μ) (∫ (x : X) in ⋃ n, s n, f x ∂μ)
-- MeasureTheory.integral_iUnion
theorem integral_iUnion {ι : Type*} [Countable ι] {s : ι → Set X} (hm : ∀ i, MeasurableSet (s i))
    (hd : Pairwise (Disjoint on s)) (hfi : IntegrableOn f (⋃ i, s i) μ) :
    ∫ x in ⋃ n, s n, f x ∂μ = ∑' n, ∫ x in s n, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] {s : ι → Set X} (hm : ∀ (i : ι), MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (hfi : MeasureTheory.IntegrableOn f (⋃ i, s i) μ)
Goal: ∫ (x : X) in ⋃ n, s n, f x ∂μ = ∑' (n : ι), ∫ (x : X) in s n, f x ∂μ
-- MeasureTheory.integral_iUnion_ae
theorem integral_iUnion_ae {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) : ∫ x in ⋃ n, s n, f x ∂μ = ∑' n, ∫ x in s n, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Countable ι] {s : ι → Set X} (hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (hfi : MeasureTheory.IntegrableOn f (⋃ i, s i) μ)
Goal: ∫ (x : X) in ⋃ n, s n, f x ∂μ = ∑' (n : ι), ∫ (x : X) in s n, f x ∂μ
-- MeasureTheory.setIntegral_eq_zero_of_ae_eq_zero
theorem setIntegral_eq_zero_of_ae_eq_zero (ht_eq : ∀ᵐ x ∂μ, x ∈ t → f x = 0) :
    ∫ x in t, f x ∂μ = 0 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {t : Set X} {μ : MeasureTheory.Measure X} (ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0)
Goal: ∫ (x : X) in t, f x ∂μ = 0
-- MeasureTheory.setIntegral_eq_zero_of_forall_eq_zero
theorem setIntegral_eq_zero_of_forall_eq_zero (ht_eq : ∀ x ∈ t, f x = 0) :
    ∫ x in t, f x ∂μ = 0 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {t : Set X} {μ : MeasureTheory.Measure X} (ht_eq : ∀ x ∈ t, f x = 0)
Goal: ∫ (x : X) in t, f x ∂μ = 0
-- MeasureTheory.integral_union_eq_left_of_ae_aux
theorem integral_union_eq_left_of_ae_aux (ht_eq : ∀ᵐ x ∂μ.restrict t, f x = 0)
    (haux : StronglyMeasurable f) (H : IntegrableOn f (s ∪ t) μ) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0) (haux : MeasureTheory.StronglyMeasurable f) (H : MeasureTheory.IntegrableOn f (s ∪ t) μ)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_union_eq_left_of_ae
theorem integral_union_eq_left_of_ae (ht_eq : ∀ᵐ x ∂μ.restrict t, f x = 0) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_union_eq_left_of_forall₀
theorem integral_union_eq_left_of_forall₀ {f : X → E} (ht : NullMeasurableSet t μ)
    (ht_eq : ∀ x ∈ t, f x = 0) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {s t : Set X} {μ : MeasureTheory.Measure X} {f : X → E} (ht : MeasureTheory.NullMeasurableSet t μ) (ht_eq : ∀ x ∈ t, f x = 0)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_union_eq_left_of_forall
theorem integral_union_eq_left_of_forall {f : X → E} (ht : MeasurableSet t)
    (ht_eq : ∀ x ∈ t, f x = 0) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {s t : Set X} {μ : MeasureTheory.Measure X} {f : X → E} (ht : MeasurableSet t) (ht_eq : ∀ x ∈ t, f x = 0)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux
theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux (hts : s ⊆ t)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) (haux : StronglyMeasurable f)
    (h'aux : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (hts : s ⊆ t) (h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0) (haux : MeasureTheory.StronglyMeasurable f) (h'aux : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_eq_of_subset_of_ae_diff_eq_zero
theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero (ht : NullMeasurableSet t μ) (hts : s ⊆ t)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasureTheory.NullMeasurableSet t μ) (hts : s ⊆ t) (h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0)
Goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_eq_of_subset_of_forall_diff_eq_zero
theorem setIntegral_eq_of_subset_of_forall_diff_eq_zero (ht : MeasurableSet t) (hts : s ⊆ t)
    (h't : ∀ x ∈ t \ s, f x = 0) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasurableSet t) (hts : s ⊆ t) (h't : ∀ x ∈ t \ s, f x = 0)
Goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.setIntegral_eq_integral_of_ae_compl_eq_zero
theorem setIntegral_eq_integral_of_ae_compl_eq_zero (h : ∀ᵐ x ∂μ, x ∉ s → f x = 0) :
    ∫ x in s, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ
-- MeasureTheory.setIntegral_eq_integral_of_forall_compl_eq_zero
theorem setIntegral_eq_integral_of_forall_compl_eq_zero (h : ∀ x, x ∉ s → f x = 0) :
    ∫ x in s, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (h : ∀ x ∉ s, f x = 0)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ
-- MeasureTheory.setIntegral_neg_eq_setIntegral_nonpos
theorem setIntegral_neg_eq_setIntegral_nonpos [LinearOrder E] {f : X → E}
    (hf : AEStronglyMeasurable f μ) :
    ∫ x in {x | f x < 0}, f x ∂μ = ∫ x in {x | f x ≤ 0}, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} [LinearOrder E] {f : X → E} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ
-- MeasureTheory.integral_norm_eq_pos_sub_neg
theorem integral_norm_eq_pos_sub_neg {f : X → ℝ} (hfi : Integrable f μ) :
    ∫ x, ‖f x‖ ∂μ = ∫ x in {x | 0 ≤ f x}, f x ∂μ - ∫ x in {x | f x ≤ 0}, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {f : X → ℝ} (hfi : MeasureTheory.Integrable f μ)
Goal: ∫ (x : X), ‖f x‖ ∂μ = ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ
-- MeasureTheory.setIntegral_const
theorem setIntegral_const [CompleteSpace E] (c : E) : ∫ _ in s, c ∂μ = (μ s).toReal • c 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set X} {μ : MeasureTheory.Measure X} [CompleteSpace E] (c : E)
Goal: ∫ (x : X) in s, c ∂μ = (μ s).toReal • c
-- MeasureTheory.integral_indicator_const
@[simp]
theorem integral_indicator_const [CompleteSpace E] (e : E) ⦃s : Set X⦄ (s_meas : MeasurableSet s) :
    ∫ x : X, s.indicator (fun _ : X => e) x ∂μ = (μ s).toReal • e 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} [CompleteSpace E] (e : E) ⦃s : Set X⦄ (s_meas : MeasurableSet s)
Goal: ∫ (x : X), s.indicator (fun x => e) x ∂μ = (μ s).toReal • e
-- MeasureTheory.integral_indicator_one
@[simp]
theorem integral_indicator_one ⦃s : Set X⦄ (hs : MeasurableSet s) :
    ∫ x, s.indicator 1 x ∂μ = (μ s).toReal 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} ⦃s : Set X⦄ (hs : MeasurableSet s)
Goal: ∫ (x : X), s.indicator 1 x ∂μ = (μ s).toReal
-- MeasureTheory.setIntegral_indicatorConstLp
theorem setIntegral_indicatorConstLp [CompleteSpace E]
    {p : ℝ≥0∞} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
    ∫ x in s, indicatorConstLp p ht hμt e x ∂μ = (μ (t ∩ s)).toReal • e 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {s t : Set X} {μ : MeasureTheory.Measure X} [CompleteSpace E] {p : ℝ≥0∞} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμt : μ t ≠ ⊤) (e : E)
Goal: ∫ (x : X) in s, ↑↑(MeasureTheory.indicatorConstLp p ht hμt e) x ∂μ = (μ (t ∩ s)).toReal • e
-- MeasureTheory.integral_indicatorConstLp
theorem integral_indicatorConstLp [CompleteSpace E]
    {p : ℝ≥0∞} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
    ∫ x, indicatorConstLp p ht hμt e x ∂μ = (μ t).toReal • e 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {t : Set X} {μ : MeasureTheory.Measure X} [CompleteSpace E] {p : ℝ≥0∞} (ht : MeasurableSet t) (hμt : μ t ≠ ⊤) (e : E)
Goal: ∫ (x : X), ↑↑(MeasureTheory.indicatorConstLp p ht hμt e) x ∂μ = (μ t).toReal • e
-- MeasureTheory.setIntegral_map
theorem setIntegral_map {Y} [MeasurableSpace Y] {g : X → Y} {f : Y → E} {s : Set Y}
    (hs : MeasurableSet s) (hf : AEStronglyMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) :
    ∫ y in s, f y ∂Measure.map g μ = ∫ x in g ⁻¹' s, f (g x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5} [MeasurableSpace Y] {g : X → Y} {f : Y → E} {s : Set Y} (hs : MeasurableSet s) (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map g μ)) (hg : AEMeasurable g μ)
Goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ
-- MeasurableEmbedding.setIntegral_map
theorem _root_.MeasurableEmbedding.setIntegral_map {Y} {_ : MeasurableSpace Y} {f : X → Y}
    (hf : MeasurableEmbedding f) (g : Y → E) (s : Set Y) :
    ∫ y in s, g y ∂Measure.map f μ = ∫ x in f ⁻¹' s, g (f x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5}
Goal: ∀ {x : MeasurableSpace Y} {f : X → Y},
  MeasurableEmbedding f →
    ∀ (g : Y → E) (s : Set Y), ∫ (y : Y) in s, g y ∂MeasureTheory.Measure.map f μ = ∫ (x : X) in f ⁻¹' s, g (f x) ∂μ
-- ClosedEmbedding.setIntegral_map
theorem _root_.ClosedEmbedding.setIntegral_map [TopologicalSpace X] [BorelSpace X] {Y}
    [MeasurableSpace Y] [TopologicalSpace Y] [BorelSpace Y] {g : X → Y} {f : Y → E} (s : Set Y)
    (hg : ClosedEmbedding g) : ∫ y in s, f y ∂Measure.map g μ = ∫ x in g ⁻¹' s, f (g x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} [TopologicalSpace X] [BorelSpace X] {Y : Type u_5} [MeasurableSpace Y] [TopologicalSpace Y] [BorelSpace Y] {g : X → Y} {f : Y → E} (s : Set Y) (hg : ClosedEmbedding g)
Goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ
-- MeasureTheory.MeasurePreserving.setIntegral_preimage_emb
theorem MeasurePreserving.setIntegral_preimage_emb {Y} {_ : MeasurableSpace Y} {f : X → Y} {ν}
    (h₁ : MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) (g : Y → E) (s : Set Y) :
    ∫ x in f ⁻¹' s, g (f x) ∂μ = ∫ y in s, g y ∂ν 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5}
Goal: ∀ {x : MeasurableSpace Y} {f : X → Y} {ν : MeasureTheory.Measure Y},
  MeasureTheory.MeasurePreserving f μ ν →
    MeasurableEmbedding f → ∀ (g : Y → E) (s : Set Y), ∫ (x : X) in f ⁻¹' s, g (f x) ∂μ = ∫ (y : Y) in s, g y ∂ν
-- MeasureTheory.MeasurePreserving.setIntegral_image_emb
theorem MeasurePreserving.setIntegral_image_emb {Y} {_ : MeasurableSpace Y} {f : X → Y} {ν}
    (h₁ : MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) (g : Y → E) (s : Set X) :
    ∫ y in f '' s, g y ∂ν = ∫ x in s, g (f x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5}
Goal: ∀ {x : MeasurableSpace Y} {f : X → Y} {ν : MeasureTheory.Measure Y},
  MeasureTheory.MeasurePreserving f μ ν →
    MeasurableEmbedding f → ∀ (g : Y → E) (s : Set X), ∫ (y : Y) in f '' s, g y ∂ν = ∫ (x : X) in s, g (f x) ∂μ
-- MeasureTheory.setIntegral_map_equiv
theorem setIntegral_map_equiv {Y} [MeasurableSpace Y] (e : X ≃ᵐ Y) (f : Y → E) (s : Set Y) :
    ∫ y in s, f y ∂Measure.map e μ = ∫ x in e ⁻¹' s, f (e x) ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5} [MeasurableSpace Y] (e : X ≃ᵐ Y) (f : Y → E) (s : Set Y)
Goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map (⇑e) μ = ∫ (x : X) in ⇑e ⁻¹' s, f (e x) ∂μ
-- MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae
theorem norm_setIntegral_le_of_norm_le_const_ae {C : ℝ} (hs : μ s < ∞)
    (hC : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} {C : ℝ} (hs : μ s < ⊤) (hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C)
Goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
-- MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae'
theorem norm_setIntegral_le_of_norm_le_const_ae' {C : ℝ} (hs : μ s < ∞)
    (hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) (hfm : AEStronglyMeasurable f (μ.restrict s)) :
    ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} {C : ℝ} (hs : μ s < ⊤) (hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C) (hfm : MeasureTheory.AEStronglyMeasurable f (μ.restrict s))
Goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
-- MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae''
theorem norm_setIntegral_le_of_norm_le_const_ae'' {C : ℝ} (hs : μ s < ∞) (hsm : MeasurableSet s)
    (hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} {C : ℝ} (hs : μ s < ⊤) (hsm : MeasurableSet s) (hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C)
Goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
-- MeasureTheory.norm_setIntegral_le_of_norm_le_const
theorem norm_setIntegral_le_of_norm_le_const {C : ℝ} (hs : μ s < ∞) (hC : ∀ x ∈ s, ‖f x‖ ≤ C)
    (hfm : AEStronglyMeasurable f (μ.restrict s)) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} {C : ℝ} (hs : μ s < ⊤) (hC : ∀ x ∈ s, ‖f x‖ ≤ C) (hfm : MeasureTheory.AEStronglyMeasurable f (μ.restrict s))
Goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
-- MeasureTheory.norm_setIntegral_le_of_norm_le_const'
theorem norm_setIntegral_le_of_norm_le_const' {C : ℝ} (hs : μ s < ∞) (hsm : MeasurableSet s)
    (hC : ∀ x ∈ s, ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : MeasureTheory.Measure X} {C : ℝ} (hs : μ s < ⊤) (hsm : MeasurableSet s) (hC : ∀ x ∈ s, ‖f x‖ ≤ C)
Goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
-- MeasureTheory.setIntegral_eq_zero_iff_of_nonneg_ae
theorem setIntegral_eq_zero_iff_of_nonneg_ae {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
    (hfi : IntegrableOn f s μ) : ∫ x in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0 
Arguments: {X : Type u_1} [MeasurableSpace X] {s : Set X} {μ : MeasureTheory.Measure X} {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f) (hfi : MeasureTheory.IntegrableOn f s μ)
Goal: ∫ (x : X) in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0
-- MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae
theorem setIntegral_pos_iff_support_of_nonneg_ae {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
    (hfi : IntegrableOn f s μ) : (0 < ∫ x in s, f x ∂μ) ↔ 0 < μ (support f ∩ s) 
Arguments: {X : Type u_1} [MeasurableSpace X] {s : Set X} {μ : MeasureTheory.Measure X} {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f) (hfi : MeasureTheory.IntegrableOn f s μ)
Goal: 0 < ∫ (x : X) in s, f x ∂μ ↔ 0 < μ (Function.support f ∩ s)
-- MeasureTheory.setIntegral_gt_gt
theorem setIntegral_gt_gt {R : ℝ} {f : X → ℝ} (hR : 0 ≤ R)
    (hfint : IntegrableOn f {x | ↑R < f x} μ) (hμ : μ {x | ↑R < f x} ≠ 0) :
    (μ {x | ↑R < f x}).toReal * R < ∫ x in {x | ↑R < f x}, f x ∂μ 
Arguments: {X : Type u_1} [MeasurableSpace X] {μ : MeasureTheory.Measure X} {R : ℝ} {f : X → ℝ} (hR : 0 ≤ R) (hfint : MeasureTheory.IntegrableOn f {x | R < f x} μ) (hμ : μ {x | R < f x} ≠ 0)
Goal: (μ {x | R < f x}).toReal * R < ∫ (x : X) in {x | R < f x}, f x ∂μ