isInvInvariant — Mathlib

English

An additive analogue asserts the invariance of the product measure under inversion on the product group across all coordinates when volumes are invariant.

Русский

Аналогично противление под инверсией на всем пространстве координат.

LaTeX

$$$\\text{IsInvInvariant}(\\mathrm{volume})$$$

Lean4

@[to_additive]
instance isInvInvariant [∀ i, Group (α i)] [∀ i, MeasurableInv (α i)] [∀ i, IsInvInvariant (μ i)] :
    IsInvInvariant (Measure.pi μ) := by
  refine ⟨(Measure.pi_eq fun s hs => ?_).symm⟩
  have A : Inv.inv ⁻¹' pi univ s = Set.pi univ fun i => Inv.inv ⁻¹' s i := by ext; simp
  simp_rw [Measure.inv, Measure.map_apply measurable_inv (MeasurableSet.univ_pi hs), A, pi_pi, measure_preimage_inv]

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