comp_inv — Mathlib

English

If volume ν on α is sigma-finite and G has a fundamental domain, then the quotient μ on α/G that satisfies QuotientMeasureEqMeasurePreimage is sigma-finite.

Русский

Если объём ν на α сигма-конечен и у G есть фундаментальная область, то мера μ на α/G, удовлетворяющая QuotientMeasureEqMeasurePreimage, сигма-конечна.

LaTeX

$$$[\\SigmaFinite \\nu][HasFundamentalDomain G α]\\Rightarrow \\SigmaFinite(μ)\\, (\\text{where } μ \\text{ satisfies quotient preimage}).$$$

Lean4

@[to_additive]
theorem comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) : Integrable (fun t => f t⁻¹) μ :=
  (hf.mono_measure (map_inv_eq_self μ).le).comp_measurable measurable_inv

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