sFinite_of_absolutelyContinuous — Mathlib

English

If μ is absolutely continuous with respect to ν and ν is s-finite, then μ is s-finite.

Русский

Если μ абсолютно непрерывна относительно ν и ν s-фини, то μ также s-фinite.

LaTeX

$$$μ ≪ ν \quad and \quad ν \, \text{is } SFinite \Rightarrow SFinite μ$$$

Lean4

/-- If `μ ≪ ν` and `ν` is s-finite, then `μ` is s-finite. -/
theorem sFinite_of_absolutelyContinuous {ν : Measure α} [SFinite ν] (hμν : μ ≪ ν) : SFinite μ :=
  by
  rw [← Measure.restrict_add_restrict_compl (μ := μ) measurableSet_sigmaFiniteSetWRT,
    restrict_compl_sigmaFiniteSetWRT hμν]
  infer_instance

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