Theorems · Theorem · measure theory

aestronglyMeasurable_smul_iff

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  {f : α → β} {G : Type u_6} [inst_1 : Group G] [inst_2 : MulAction G β] [inst_3 : TopologicalSpace G] [ContinuousInv G]
  [ContinuousSMul G β] {c : α → G},
  MeasureTheory.AEStronglyMeasurable c μ →
    (MeasureTheory.AEStronglyMeasurable (fun x => c x • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ)

Multiplying by an a.e. strongly measurable scalar function with values in a group preserves a.e. strong measurability. This is the varying-scalar analogue of aestronglyMeasurable_const_smul_iff.

Defined in

Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Cited by

0 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceGroupMulActionTopologicalSpaceContinuousInvContinuousSMul