Theorems · Theorem · measure theory

MeasureTheory.exists_eq_setAverage

∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [inst : TopologicalSpace α]
  {f : α → ℝ},
  IsConnected s →
    ContinuousOn f s → MeasureTheory.IntegrableOn f s μ → μ s ≠ ⊤ → μ s ≠ 0 → ∃ c ∈ s, f c = ⨍ (x : α) in s, f x ∂μ

If s is a connected set of finite, nonzero μ-measure and f : α → ℝ is continuous on s and integrable on s w.r.t. μ, then f attains its μ-average on s.

Defined in

Mathlib.MeasureTheory.Integral.Average

Cited by

3 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpace