Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le

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

Given a simple function f with values in ℝ≥0, there exists an upper 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