Theorems · Theorem · measure theory

MeasureTheory.Measure.measure_toMeasurable_inter_of_sum

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},
  MeasurableSet s →
    ∀ {t : Set α} {m : ℕ → MeasureTheory.Measure α},
      (∀ (n : ℕ), (m n) t ≠ ⊤) → μ = MeasureTheory.Measure.sum m → μ (MeasureTheory.toMeasurable μ t ∩ s) = μ (t ∩ s)

If a measure μ is the sum of a countable family mₙ, and a set t has finite measure for each mₙ, then its measurable superset toMeasurable μ t (which has the same measure as t) satisfies, for any measurable set s, the equality μ (toMeasurable μ t ∩ s) = μ (t ∩ s).

Defined in

Mathlib.MeasureTheory.Measure.Typeclasses.SFinite

Cited by

1 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound