{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = Finset.sum Finset.univ fun x => ↑↑μ {(x, y)}"}
{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = ∑' (x : α), ↑↑μ {(x, y)}"}
{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = ∑' (y : β), ↑↑μ {(x, y)}"}
{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = Finset.sum Finset.univ fun y => ↑↑μ {(x, y)}"}