measurePreserving_prod_inv_mul — Mathlib

English

The inverse map Inv.inv is measurable and quasi-absolutely continuous with respect to μ; i.e., μ is quasi-xy absolutely continuous under inversion.

Русский

Обратная карта инвариантна к измеримости и к Absolutely continuous относительно μ; и т.д.

LaTeX

$$QuasiMeasurePreserving Inv.inv μ μ$$

Lean4

/-- The map `(x, y) ↦ (x, x⁻¹y)` is measure-preserving.
This is the function `S⁻¹` in [Halmos, §59],
where `S` is the map `(x, y) ↦ (x, xy)`. -/
@[to_additive measurePreserving_prod_neg_add /-- The map `(x, y) ↦ (x, - x + y)` is measure-preserving. -/
    ]
theorem measurePreserving_prod_inv_mul [IsMulLeftInvariant ν] :
    MeasurePreserving (fun z : G × G => (z.1, z.1⁻¹ * z.2)) (μ.prod ν) (μ.prod ν) :=
  (measurePreserving_prod_mul μ ν).symm <| MeasurableEquiv.shearMulRight G

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