domSMul_apply — Mathlib

English

For any g₀ ≠ 0, (mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg.

Русский

Для любого g hg ≠ 0, (mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg.

LaTeX

$$${(\\mathrm{mulLeft}_{0} g\\, hg).toEquiv = \\mathrm{Equiv.mulLeft}_{0} g\\, hg}$$$

Lean4

theorem domSMul_apply (μ : Measure A) (g : Gᵈᵐᵃ) (s : Set A) : (g • μ) s = μ (DomMulAct.mk.symm g • s) :=
  by
  refine ((MeasurableEquiv.smul ((DomMulAct.mk.symm g : G)⁻¹)).map_apply _).trans ?_
  congr 1
  exact Set.preimage_smul_inv (DomMulAct.mk.symm g) s

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