measure_inter_eq_zero_of_restrict

English

For measurable s, (μ|_s)(t) = 0 iff μ(t ∩ s) = 0; equivalently, a certain Iff characterization of restricted measure vanishes.

Русский

Для измеримого s, (μ|_s)(t) = 0 эквивалентно μ(t ∩ s) = 0; эквивалентное характеризация нуля ограниченной меры.

LaTeX

$$$ (\mu \restriction s)(t) = 0 \iff \mu(t \cap s) = 0 $ for MeasurableSet s$$

Lean4

theorem measure_inter_eq_zero_of_restrict (h : μ.restrict s t = 0) : μ (t ∩ s) = 0 :=
  nonpos_iff_eq_zero.1 (h ▸ le_restrict_apply _ _)

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