measure_eq_univ_iff_eq — Mathlib

English

Let F be closed and μ a finite measure on an opens-measurable space. Then μ(F) = μ(univ) if and only if F = univ.

Русский

Пусть F замкнуто и μ — конечно-мерная мера на пространстве с открытым измеримым пространством. Тогда μ(F) = μ(univ) эквивалентно F = univ.

LaTeX

$$$\\forall F\\ [\\text{OpensMeasurableSpace }X]\\ [\\text{Measure } μ],\\; \\text{IsFiniteMeasure } μ,\\; IsClosed F \\rightarrow (μ(F) = μ(\\mathrm{univ}) \\iff F = \\mathrm{univ}).$$$

Lean4

theorem _root_.IsClosed.measure_eq_univ_iff_eq [OpensMeasurableSpace X] [IsFiniteMeasure μ] (hF : IsClosed F) :
    μ F = μ univ ↔ F = univ := by
  rw [← ae_eq_univ_iff_measure_eq hF.measurableSet.nullMeasurableSet, hF.ae_eq_univ_iff_eq]

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