Theorems · Theorem · measure theory

MeasureTheory.Measure.exists_innerRegular_eq_of_isCompact

∀ {X : Type u_1} [inst : TopologicalSpace X] [T2Space X] [inst_2 : MeasurableSpace X] [BorelSpace X]
  (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] {K : Set X},
  IsCompact K →
    μ Kᶜ = 0 →
      ∃ ν,
        ν.InnerRegular ∧
          MeasureTheory.IsFiniteMeasure ν ∧
            ν Kᶜ = 0 ∧ ∀ (g : BoundedContinuousFunction X ℝ), ∫ (x : X), g x ∂μ = ∫ (x : X), g x ∂ν

Given a finite measure supported on a compact set, there exists another finite measure which integrates in the same way bounded continuous functions, and is regular.

Defined in

Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Cited by

1 results in Mathlib

Foundations

Depth 263 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceT2SpaceMeasurableSpaceBorelSpaceMeasureTheory.IsFiniteMeasure