English
If μ is left-invariant on G and f: G →ₙ* H is a surjective measurable morphism, then the pushforward measure map f μ is left-invariant on H.
Русский
Если μ инвариантна слева на G и f: G →ₙ* H десяксимерная сюръекция, то образ меры f μ по f остаётся слева инвариантной на H.
LaTeX
$$$\forall f: G \to* H,\ hf: Measurable f,\ hsurj: Surjective f,\ IsMulLeftInvariant(μ) \Rightarrow IsMulLeftInvariant(Measure.map f μ)$$$
Lean4
@[to_additive]
theorem isMulLeftInvariant_map {H : Type*} [MeasurableSpace H] [Mul H] [MeasurableMul H] [IsMulLeftInvariant μ]
(f : G →ₙ* H) (hf : Measurable f) (h_surj : Surjective f) : IsMulLeftInvariant (Measure.map f μ) :=
by
refine ⟨fun h => ?_⟩
rw [map_map (measurable_const_mul _) hf]
obtain ⟨g, rfl⟩ := h_surj h
conv_rhs => rw [← map_mul_left_eq_self μ g]
rw [map_map hf (measurable_const_mul _)]
congr 2
ext y
simp only [comp_apply, map_mul]
