comp — Mathlib

English

If μ ≪ ν and for μ-a.e. a we have κ(a) ≪ η(a), then κ ∘ₘ μ ≪ η ∘ₘ ν.

Русский

Если μ ⪯ ν и для μ-п Almost sure κ(a) ≪ η(a), то κ ∘ μ ⪯ η ∘ ν.

LaTeX

$$$\\mu \\ll \\nu \\Rightarrow (\\kappa \\circ_{m} \\mu) \\ll (\\eta \\circ_{m} \\nu)$$$

Lean4

theorem comp (hμν : μ ≪ ν) (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) : κ ∘ₘ μ ≪ η ∘ₘ ν :=
  (AbsolutelyContinuous.comp_left μ hκη).trans (hμν.comp_right η)

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