Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge

∀ {α : Type u_1} [inst : TopologicalSpace α] [inst_1 : MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α)
  [μ.WeaklyRegular] (f : MeasureTheory.SimpleFunc α NNReal) {ε : ENNReal},
  ε ≠ 0 → ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Given a simple function f with values in ℝ≥0, there exists a lower semicontinuous function g ≥ f with integral arbitrarily close to that of f. Formulation in terms of lintegral. Auxiliary lemma for Vitali-Carathéodory theorem exists_lt_lower_semicontinuous_integral_lt.

Defined in

Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.WeaklyRegular