Theorems · Theorem · measure theory

MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top

∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α}
  [OpensMeasurableSpace α] [μ.InnerRegularCompactLTTop],
  MeasurableSet s → μ s ≠ ⊤ → μ.IsEverywherePos (μ.everywherePosSubset s)

In a space with an inner regular measure for finite measure sets, the everywhere positive subset of a measurable set of finite measure is itself everywhere positive. This is not obvious as μ.everywherePosSubset s is defined as the points whose neighborhoods intersect s along positive measure subsets, but this does not say they also intersect μ.everywherePosSubset s along positive measure subsets.

Defined in

Mathlib.MeasureTheory.Measure.EverywherePos

Cited by

2 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMeasurableSpaceOpensMeasurableSpaceMeasureTheory.Measure.InnerRegularCompactLTTop