Theorems · Theorem · measure theory

MeasureTheory.MeasurePreserving.prod

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β]
  [inst_2 : MeasurableSpace γ] {δ : Type u_4} [inst_3 : MeasurableSpace δ] {μa : MeasureTheory.Measure α}
  {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {μd : MeasureTheory.Measure δ}
  [MeasureTheory.SFinite μa] [MeasureTheory.SFinite μc] {f : α → β} {g : γ → δ},
  MeasureTheory.MeasurePreserving f μa μb →
    MeasureTheory.MeasurePreserving g μc μd → MeasureTheory.MeasurePreserving (Prod.map f g) (μa.prod μc) (μb.prod μd)

If f : α → β sends the measure μa to μb and g : γ → δ sends the measure μc to μd, then Prod.map f g sends μa.prod μc to μb.prod μd.

Defined in

Mathlib.MeasureTheory.Measure.Prod

Cited by

4 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceMeasurableSpaceMeasurableSpaceMeasureTheory.SFiniteMeasureTheory.SFinite