| {"name":"MeasureTheory.lintegral_eq_sum'","declaration":"theorem MeasureTheory.lintegral_eq_sum' {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) {s : Finset α} (hA : ↑↑μ (↑s)ᶜ = 0) (f : α → ENNReal) : ∫⁻ (x : α), f x ∂μ = Finset.sum s fun x => f x * ↑↑μ {x}"} | |
| {"name":"MeasureTheory.lintegral_eq_single","declaration":"theorem MeasureTheory.lintegral_eq_single {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (a : α) (f : α → ENNReal) (ha : ∀ (b : α), b ≠ a → f b = 0) : ∫⁻ (x : α), f x ∂μ = f a * ↑↑μ {a}"} | |
| {"name":"MeasureTheory.lintegral_eq_sum","declaration":"theorem MeasureTheory.lintegral_eq_sum {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Fintype α] : ∫⁻ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => ↑↑μ {x} * f x"} | |
| {"name":"MeasureTheory.lintegral_eq_zero_of_ae_zero","declaration":"theorem MeasureTheory.lintegral_eq_zero_of_ae_zero {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ENNReal} {E : Set α} (hE : ↑↑μ Eᶜ = 0) (hf : ∀ x ∈ E, f x = 0) (hmes : MeasurableSet E) : ∫⁻ (x : α), f x ∂μ = 0"} | |
| {"name":"MeasureTheory.lintegral_eq_sum_countable","declaration":"theorem MeasureTheory.lintegral_eq_sum_countable {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Countable α] : ∫⁻ (x : α), f x ∂μ = ∑' (x : α), ↑↑μ {x} * f x"} | |