English
If a measure μ on G is Haar and μ is finite on compacts, then for any equivariant map and any n, the z-pow preservation property holds.
Русский
Если μ — мера Хаара и конечна на компактах, то при любом эквивариантном отображении сохраняется z-повоедение.
LaTeX
$$$\\forall G \\; [CommGroup G] \\; [TopologicalSpace G] \\; [IsTopologicalGroup G] \\; [MeasurableSpace G] \\; [BorelSpace G] \\; (μ : \\mathrm{Measure}(G)) \\; [IsHaarMeasure μ] \\, {n : \\mathbb{Z}} \\, (hn : n \\neq 0) \\\\Rightarrow \\text{MeasurePreserving } (g \\mapsto g^n) \\ μ \\ μ$$
Lean4
/-- If a sequence of bounded continuous functions tends to the indicator of a measurable set and
the functions are uniformly bounded, then their integrals against a finite measure tend to the
measure of the set.
A similar result with more general assumptions is
`MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator`.
-/
theorem measure_of_cont_bdd_of_tendsto_indicator (μ : Measure Ω) [IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω}
(E_mble : MeasurableSet E) (fs : ℕ → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ n ω, fs n ω ≤ c)
(fs_lim : Tendsto (fun n ω ↦ fs n ω) atTop (𝓝 (indicator E fun _ ↦ (1 : ℝ≥0)))) :
Tendsto (fun n ↦ lintegral μ fun ω ↦ fs n ω) atTop (𝓝 (μ E)) :=
by
have fs_lim' : ∀ ω, Tendsto (fun n : ℕ ↦ (fs n ω : ℝ≥0)) atTop (𝓝 (indicator E (fun _ ↦ (1 : ℝ≥0)) ω)) :=
by
rw [tendsto_pi_nhds] at fs_lim
exact fun ω ↦ fs_lim ω
apply
measure_of_cont_bdd_of_tendsto_filter_indicator μ E_mble fs
(Eventually.of_forall fun n ↦ Eventually.of_forall (fs_bdd n)) (Eventually.of_forall fs_lim')
