ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.limsup_lintegral_le
theorem limsup_lintegral_le {f : ℕ → α → ℝ≥0∞} (g : α → ℝ≥0∞) (hf_meas : ∀ n, Measurable (f n))
    (h_bound : ∀ n, f n ≤ᵐ[μ] g) (h_fin : ∫⁻ a, g a ∂μ ≠ ∞) :
    limsup (fun n => ∫⁻ a, f n a ∂μ) atTop ≤ ∫⁻ a, limsup (fun n => f n a) atTop ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} (g : α → ℝ≥0∞) (hf_meas : ∀ (n : ℕ), Measurable (f n)) (h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g) (h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤)
Goal: Filter.limsup (fun n => ∫⁻ (a : α), f n a ∂μ) Filter.atTop ≤ ∫⁻ (a : α), Filter.limsup (fun n => f n a) Filter.atTop ∂μ
-- MeasureTheory.tendsto_lintegral_of_dominated_convergence
theorem tendsto_lintegral_of_dominated_convergence {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞}
    (bound : α → ℝ≥0∞) (hF_meas : ∀ n, Measurable (F n)) (h_bound : ∀ n, F n ≤ᵐ[μ] bound)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ (n : ℕ), Measurable (F n)) (h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
-- MeasureTheory.tendsto_lintegral_of_dominated_convergence'
theorem tendsto_lintegral_of_dominated_convergence' {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞}
    (bound : α → ℝ≥0∞) (hF_meas : ∀ n, AEMeasurable (F n) μ) (h_bound : ∀ n, F n ≤ᵐ[μ] bound)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) atTop (𝓝 (∫⁻ a, f a ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {F : ℕ → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ) (h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
-- MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence
theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι}
    [l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞)
    (hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a)
    (h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :
    Tendsto (fun n => ∫⁻ a, F n a ∂μ) l (𝓝 <| ∫⁻ a, f a ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} {l : Filter ι} [l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞) (hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)) (h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) (h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) l (𝓝 (f a)))
Goal: Filter.Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) l (𝓝 (∫⁻ (a : α), f a ∂μ))
-- MeasureTheory.lintegral_tendsto_of_tendsto_of_antitone
theorem lintegral_tendsto_of_tendsto_of_antitone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
    (hf : ∀ n, AEMeasurable (f n) μ) (h_anti : ∀ᵐ x ∂μ, Antitone fun n ↦ f n x)
    (h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))) :
    Tendsto (fun n ↦ ∫⁻ x, f n x ∂μ) atTop (𝓝 (∫⁻ x, F x ∂μ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} (hf : ∀ (n : ℕ), AEMeasurable (f n) μ) (h_anti : ∀ᵐ (x : α) ∂μ, Antitone fun n => f n x) (h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤) (h_tendsto : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (F x)))
Goal: Filter.Tendsto (fun n => ∫⁻ (x : α), f n x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), F x ∂μ))
-- MeasureTheory.lintegral_iSup_directed_of_measurable
theorem lintegral_iSup_directed_of_measurable [Countable β] {f : β → α → ℝ≥0∞}
    (hf : ∀ b, Measurable (f b)) (h_directed : Directed (· ≤ ·) f) :
    ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ (b : β), Measurable (f b)) (h_directed : Directed (fun x x_1 => x ≤ x_1) f)
Goal: ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- MeasureTheory.lintegral_iSup_directed
theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
    (h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ (b : β), AEMeasurable (f b) μ) (h_directed : Directed (fun x x_1 => x ≤ x_1) f)
Goal: ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- MeasureTheory.lintegral_tsum
theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) :
    ∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ (i : β), AEMeasurable (f i) μ)
Goal: ∫⁻ (a : α), ∑' (i : β), f i a ∂μ = ∑' (i : β), ∫⁻ (a : α), f i a ∂μ
-- MeasureTheory.lintegral_iUnion₀
theorem lintegral_iUnion₀ [Countable β] {s : β → Set α} (hm : ∀ i, NullMeasurableSet (s i) μ)
    (hd : Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {s : β → Set α} (hm : ∀ (i : β), MeasureTheory.NullMeasurableSet (s i) μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ
-- MeasureTheory.lintegral_iUnion
theorem lintegral_iUnion [Countable β] {s : β → Set α} (hm : ∀ i, MeasurableSet (s i))
    (hd : Pairwise (Disjoint on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ = ∑' i, ∫⁻ a in s i, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {s : β → Set α} (hm : ∀ (i : β), MeasurableSet (s i)) (hd : Pairwise (Disjoint on s)) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ
-- MeasureTheory.lintegral_biUnion₀
theorem lintegral_biUnion₀ {t : Set β} {s : β → Set α} (ht : t.Countable)
    (hm : ∀ i ∈ t, NullMeasurableSet (s i) μ) (hd : t.Pairwise (AEDisjoint μ on s)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, MeasureTheory.NullMeasurableSet (s i) μ) (hd : t.Pairwise (MeasureTheory.AEDisjoint μ on s)) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ
-- MeasureTheory.lintegral_biUnion
theorem lintegral_biUnion {t : Set β} {s : β → Set α} (ht : t.Countable)
    (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i ∈ t, s i, f a ∂μ = ∑' i : t, ∫⁻ a in s i, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set β} {s : β → Set α} (ht : t.Countable) (hm : ∀ i ∈ t, MeasurableSet (s i)) (hd : t.PairwiseDisjoint s) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ
-- MeasureTheory.lintegral_biUnion_finset₀
theorem lintegral_biUnion_finset₀ {s : Finset β} {t : β → Set α}
    (hd : Set.Pairwise (↑s) (AEDisjoint μ on t)) (hm : ∀ b ∈ s, NullMeasurableSet (t b) μ)
    (f : α → ℝ≥0∞) : ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α} (hd : (↑s).Pairwise (MeasureTheory.AEDisjoint μ on t)) (hm : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (t b) μ) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ (a : α) in t b, f a ∂μ
-- MeasureTheory.lintegral_biUnion_finset
theorem lintegral_biUnion_finset {s : Finset β} {t : β → Set α} (hd : Set.PairwiseDisjoint (↑s) t)
    (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ a in t b, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α} (hd : (↑s).PairwiseDisjoint t) (hm : ∀ b ∈ s, MeasurableSet (t b)) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b ∈ s, ∫⁻ (a : α) in t b, f a ∂μ
-- MeasureTheory.lintegral_iUnion_le
theorem lintegral_iUnion_le [Countable β] (s : β → Set α) (f : α → ℝ≥0∞) :
    ∫⁻ a in ⋃ i, s i, f a ∂μ ≤ ∑' i, ∫⁻ a in s i, f a ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] (s : β → Set α) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ
-- MeasureTheory.lintegral_union
theorem lintegral_union {f : α → ℝ≥0∞} {A B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B) :
    ∫⁻ a in A ∪ B, f a ∂μ = ∫⁻ a in A, f a ∂μ + ∫⁻ a in B, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {A B : Set α} (hB : MeasurableSet B) (hAB : Disjoint A B)
Goal: ∫⁻ (a : α) in A ∪ B, f a ∂μ = ∫⁻ (a : α) in A, f a ∂μ + ∫⁻ (a : α) in B, f a ∂μ
-- MeasureTheory.lintegral_union_le
theorem lintegral_union_le (f : α → ℝ≥0∞) (s t : Set α) :
    ∫⁻ a in s ∪ t, f a ∂μ ≤ ∫⁻ a in s, f a ∂μ + ∫⁻ a in t, f a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) (s t : Set α)
Goal: ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in t, f a ∂μ
-- MeasureTheory.lintegral_inter_add_diff
theorem lintegral_inter_add_diff {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B) :
    ∫⁻ x in A ∩ B, f x ∂μ + ∫⁻ x in A \ B, f x ∂μ = ∫⁻ x in A, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B)
Goal: ∫⁻ (x : α) in A ∩ B, f x ∂μ + ∫⁻ (x : α) in A \ B, f x ∂μ = ∫⁻ (x : α) in A, f x ∂μ
-- MeasureTheory.lintegral_add_compl
theorem lintegral_add_compl (f : α → ℝ≥0∞) {A : Set α} (hA : MeasurableSet A) :
    ∫⁻ x in A, f x ∂μ + ∫⁻ x in Aᶜ, f x ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {A : Set α} (hA : MeasurableSet A)
Goal: ∫⁻ (x : α) in A, f x ∂μ + ∫⁻ (x : α) in Aᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLintegral_compl
theorem setLintegral_compl {f : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s)
    (hfs : ∫⁻ x in s, f x ∂μ ≠ ∞) :
    ∫⁻ x in sᶜ, f x ∂μ = ∫⁻ x, f x ∂μ - ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s) (hfs : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤)
Goal: ∫⁻ (x : α) in sᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.setLIntegral_iUnion_of_directed
theorem setLIntegral_iUnion_of_directed {ι : Type*} [Countable ι]
    (f : α → ℝ≥0∞) {s : ι → Set α} (hd : Directed (· ⊆ ·) s) :
    ∫⁻ x in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ x in s i, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} [Countable ι] (f : α → ℝ≥0∞) {s : ι → Set α} (hd : Directed (fun x x_1 => x ⊆ x_1) s)
Goal: ∫⁻ (x : α) in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ (x : α) in s i, f x ∂μ
-- MeasureTheory.lintegral_max
theorem lintegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ x, max (f x) (g x) ∂μ =
      ∫⁻ x in { x | f x ≤ g x }, g x ∂μ + ∫⁻ x in { x | g x < f x }, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
Goal: ∫⁻ (x : α), max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ
-- MeasureTheory.setLIntegral_max
theorem setLIntegral_max {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (s : Set α) :
    ∫⁻ x in s, max (f x) (g x) ∂μ =
      ∫⁻ x in s ∩ { x | f x ≤ g x }, g x ∂μ + ∫⁻ x in s ∩ { x | g x < f x }, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) (s : Set α)
Goal: ∫⁻ (x : α) in s, max (f x) (g x) ∂μ =
  ∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ
-- MeasureTheory.lintegral_map
theorem lintegral_map {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f)
    (hg : Measurable g) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g)
Goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.lintegral_map'
theorem lintegral_map' {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β}
    (hf : AEMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) :
    ∫⁻ a, f a ∂Measure.map g μ = ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : β → ℝ≥0∞} {g : α → β} (hf : AEMeasurable f (MeasureTheory.Measure.map g μ)) (hg : AEMeasurable g μ)
Goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.lintegral_map_le
theorem lintegral_map_le {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : α → β} (hg : Measurable g) :
    ∫⁻ a, f a ∂Measure.map g μ ≤ ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} (f : β → ℝ≥0∞) {g : α → β} (hg : Measurable g)
Goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ ≤ ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.lintegral_comp
theorem lintegral_comp [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f)
    (hg : Measurable g) : lintegral μ (f ∘ g) = ∫⁻ a, f a ∂map g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} (hf : Measurable f) (hg : Measurable g)
Goal: MeasureTheory.lintegral μ (f ∘ g) = ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ
-- MeasureTheory.setLIntegral_map
theorem setLIntegral_map [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β}
    (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) :
    ∫⁻ y in s, f y ∂map g μ = ∫⁻ x in g ⁻¹' s, f (g x) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β} (hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g)
Goal: ∫⁻ (y : β) in s, f y ∂MeasureTheory.Measure.map g μ = ∫⁻ (x : α) in g ⁻¹' s, f (g x) ∂μ
-- MeasureTheory.lintegral_indicator_const_comp
theorem lintegral_indicator_const_comp {mβ : MeasurableSpace β} {f : α → β} {s : Set β}
    (hf : Measurable f) (hs : MeasurableSet s) (c : ℝ≥0∞) :
    ∫⁻ a, s.indicator (fun _ => c) (f a) ∂μ = c * μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : α → β} {s : Set β} (hf : Measurable f) (hs : MeasurableSet s) (c : ℝ≥0∞)
Goal: ∫⁻ (a : α), s.indicator (fun x => c) (f a) ∂μ = c * μ (f ⁻¹' s)
-- MeasurableEmbedding.lintegral_map
theorem _root_.MeasurableEmbedding.lintegral_map [MeasurableSpace β] {g : α → β}
    (hg : MeasurableEmbedding g) (f : β → ℝ≥0∞) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {g : α → β} (hg : MeasurableEmbedding g) (f : β → ℝ≥0∞)
Goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.lintegral_map_equiv
theorem lintegral_map_equiv [MeasurableSpace β] (f : β → ℝ≥0∞) (g : α ≃ᵐ β) :
    ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] (f : β → ℝ≥0∞) (g : α ≃ᵐ β)
Goal: ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map (⇑g) μ = ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.MeasurePreserving.lintegral_map_equiv
protected theorem MeasurePreserving.lintegral_map_equiv [MeasurableSpace β] {ν : Measure β}
    (f : β → ℝ≥0∞) (g : α ≃ᵐ β) (hg : MeasurePreserving g μ ν) :
    ∫⁻ a, f a ∂ν = ∫⁻ a, f (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {ν : MeasureTheory.Measure β} (f : β → ℝ≥0∞) (g : α ≃ᵐ β) (hg : MeasureTheory.MeasurePreserving (⇑g) μ ν)
Goal: ∫⁻ (a : β), f a ∂ν = ∫⁻ (a : α), f (g a) ∂μ
-- MeasureTheory.MeasurePreserving.lintegral_comp
theorem MeasurePreserving.lintegral_comp {mb : MeasurableSpace β} {ν : Measure β} {g : α → β}
    (hg : MeasurePreserving g μ ν) {f : β → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ ν) {f : β → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν
-- MeasureTheory.MeasurePreserving.lintegral_comp_emb
theorem MeasurePreserving.lintegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β} {g : α → β}
    (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) :
    ∫⁻ a, f (g a) ∂μ = ∫⁻ b, f b ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
Goal: ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν
-- MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage
theorem MeasurePreserving.setLIntegral_comp_preimage {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞}
    (hf : Measurable f) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν
-- MeasureTheory.MeasurePreserving.setLIntegral_comp_preimage_emb
theorem MeasurePreserving.setLIntegral_comp_preimage_emb {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
    (s : Set β) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) (s : Set β)
Goal: ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν
-- MeasureTheory.MeasurePreserving.setLIntegral_comp_emb
theorem MeasurePreserving.setLIntegral_comp_emb {mb : MeasurableSpace β} {ν : Measure β}
    {g : α → β} (hg : MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞)
    (s : Set α) : ∫⁻ a in s, f (g a) ∂μ = ∫⁻ b in g '' s, f b ∂ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mb : MeasurableSpace β} {ν : MeasureTheory.Measure β} {g : α → β} (hg : MeasureTheory.MeasurePreserving g μ ν) (hge : MeasurableEmbedding g) (f : β → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (a : α) in s, f (g a) ∂μ = ∫⁻ (b : β) in g '' s, f b ∂ν
-- MeasureTheory.lintegral_subtype_comap
theorem lintegral_subtype_comap {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) :
    ∫⁻ x : s, f x ∂(μ.comap (↑)) = ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : ↑s), f ↑x ∂MeasureTheory.Measure.comap Subtype.val μ = ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.setLIntegral_subtype
theorem setLIntegral_subtype {s : Set α} (hs : MeasurableSet s) (t : Set s) (f : α → ℝ≥0∞) :
    ∫⁻ x in t, f x ∂(μ.comap (↑)) = ∫⁻ x in (↑) '' t, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (t : Set ↑s) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : ↑s) in t, f ↑x ∂MeasureTheory.Measure.comap Subtype.val μ = ∫⁻ (x : α) in Subtype.val '' t, f x ∂μ
-- MeasureTheory.exists_setLintegral_compl_lt
theorem exists_setLintegral_compl_lt {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ s : Set α, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ a in sᶜ, f a ∂μ < ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ s, MeasurableSet s ∧ μ s < ⊤ ∧ ∫⁻ (a : α) in sᶜ, f a ∂μ < ε
-- MeasureTheory.exists_measurable_le_setLintegral_eq_of_integrable
theorem exists_measurable_le_setLintegral_eq_of_integrable {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
    ∃ (g : α → ℝ≥0∞), Measurable g ∧ g ≤ f ∧ ∀ s : Set α, MeasurableSet s →
      ∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), MeasurableSet s → ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
-- MeasureTheory.lintegral_dirac'
theorem lintegral_dirac' (a : α) {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂dirac a = f a 
Arguments: {α : Type u_1} [MeasurableSpace α] (a : α) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a
-- MeasureTheory.lintegral_dirac
theorem lintegral_dirac [MeasurableSingletonClass α] (a : α) (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂dirac a = f a 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.dirac a = f a
-- MeasureTheory.setLIntegral_dirac'
theorem setLIntegral_dirac' {a : α} {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α}
    (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    ∫⁻ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {a : α} {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)]
Goal: ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0
-- MeasureTheory.setLIntegral_dirac
theorem setLIntegral_dirac {a : α} (f : α → ℝ≥0∞) (s : Set α) [MeasurableSingletonClass α]
    [Decidable (a ∈ s)] : ∫⁻ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {a : α} (f : α → ℝ≥0∞) (s : Set α) [MeasurableSingletonClass α] [Decidable (a ∈ s)]
Goal: ∫⁻ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0
-- MeasureTheory.lintegral_count'
theorem lintegral_count' {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂count = ∑' a, f a 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a
-- MeasureTheory.lintegral_count
theorem lintegral_count [MeasurableSingletonClass α] (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂count = ∑' a, f a 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂MeasureTheory.Measure.count = ∑' (a : α), f a
-- ENNReal.tsum_const_eq
theorem _root_.ENNReal.tsum_const_eq [MeasurableSingletonClass α] (c : ℝ≥0∞) :
    ∑' _ : α, c = c * Measure.count (univ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (c : ℝ≥0∞)
Goal: ∑' (x : α), c = c * MeasureTheory.Measure.count Set.univ