mutually_singular_measure_sub — Mathlib

English

Let μ and ν be finite measures on a measurable space X. Then the signed measures μ − ν and ν − μ are mutually singular.

Русский

Пусть μ и ν — конечные меры на измеримого пространства X. Тогда знаковые меры μ − ν и ν − μ взаимно сингулярны.

LaTeX

$$$ (\mu - \nu) \perp (\nu - \mu) $$$

Lean4

theorem mutually_singular_measure_sub : (μ - ν).MutuallySingular (ν - μ) :=
  by
  obtain ⟨s, hs⟩ := exists_isHahnDecomposition μ ν
  exact
    ⟨s, hs.measurableSet, sub_apply_eq_zero_of_isHahnDecomposition hs,
      sub_apply_eq_zero_of_isHahnDecomposition hs.compl⟩

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