Theorems · Theorem · measure theory

MeasureTheory.lintegral_eq_zero_of_isAddLeftInvariant

∀ {G : Type u_1} [inst : MeasurableSpace G] {μ : MeasureTheory.Measure G} [inst_1 : TopologicalSpace G]
  [inst_2 : AddGroup G] [IsTopologicalAddGroup G] [BorelSpace G] [μ.IsAddLeftInvariant] [μ.Regular] [NeZero μ]
  {f : G → ENNReal}, Continuous f → (∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0)

For nonzero regular left invariant measures, the integral of a continuous nonnegative function f is 0 iff f is 0.

Defined in

Mathlib.MeasureTheory.Group.LIntegral

Cited by

0 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceAddGroupIsTopologicalAddGroupBorelSpaceMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.Measure.RegularNeZero