Theorems · Theorem · measure theory

MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite

∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] {ε : ENNReal},
  ε ≠ 0 → ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε

In a sigma-finite measure space, there exists an integrable function which is positive everywhere (and with an arbitrarily small integral).

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Countable

Cited by

1 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceMeasureTheory.SigmaFinite