Absolute continuity on a countable type #
On a countable type with measurable singletons, absolute continuity is checked on atoms.
[UPSTREAM] candidate for Mathlib/MeasureTheory/Measure/AbsolutelyContinuous.lean.
theorem
MeasureTheory.Measure.absolutelyContinuous_of_forall_singleton
{α : Type u_1}
[MeasurableSpace α]
[Countable α]
{μ ν : Measure α}
(h : ∀ (a : α), ν {a} = 0 → μ {a} = 0)
:
μ.AbsolutelyContinuous ν
theorem
MeasureTheory.Measure.absolutelyContinuous_iff_forall_singleton
{α : Type u_1}
[MeasurableSpace α]
[Countable α]
{μ ν : Measure α}
:
μ.AbsolutelyContinuous ν ↔ ∀ (a : α), ν {a} = 0 → μ {a} = 0