integral_comp_inv_smul — Mathlib

English

Set of results about transitivity of Haar scaling under composition.

Русский

Результаты о-транзитивности масштабирования Хаара при композиции.

LaTeX

$$$mulEquivHaarChar(ψ.trans φ) = mulEquivHaarChar ψ · mulEquivHaarChar φ$$$

Lean4

/-- The integral of `f (R⁻¹ • x)` with respect to an additive Haar measure is a multiple of the
integral of `f`. The formula we give works even when `f` is not integrable or `R = 0`
thanks to the convention that a non-integrable function has integral zero. -/
theorem integral_comp_inv_smul (f : E → F) (R : ℝ) : ∫ x, f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ x, f x ∂μ := by
  rw [integral_comp_smul μ f R⁻¹, inv_pow, inv_inv]

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