Theorems · Theorem · measure theory

MeasureTheory.Integrable.measure_le_lt_top

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {f : α → β} [inst_1 : Lattice β] [HasSolidNorm β] [AddLeftMono β],
  MeasureTheory.Integrable f μ → ∀ {c : β}, c < 0 → μ {a | f a ≤ c} < ⊤

If f is integrable, then for any c < 0 the set {x | f x ≤ c} has finite measure.

Defined in

Mathlib.MeasureTheory.Function.L1Space.Integrable

Cited by

2 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupLatticeHasSolidNormAddLeftMono