English
For measurable s, the measure equality in the WRT' framework holds, linking ν s and μ s with sigma-finite restrictions.
Русский
Для измеримого s выполняется равенство по отношению ν s и μ s в рамках WRT', связывающее сигма-финитность ограничений μ|s.
LaTeX
$$$$\text{If } s\text{ is measurable},\ ν(s)\neq 0 \Rightarrow μ(s)=∞ \text{ under the}
\text{WRT' framework}.$$$
Lean4
theorem restrict_compl_sigmaFiniteSetWRT [SFinite ν] (hμν : μ ≪ ν) :
μ.restrict (μ.sigmaFiniteSetWRT ν)ᶜ = ∞ • ν.restrict (μ.sigmaFiniteSetWRT ν)ᶜ :=
by
ext s
rw [Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT.compl, Measure.smul_apply, smul_eq_mul,
Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT.compl]
by_cases hνs : ν (s ∩ (μ.sigmaFiniteSetWRT ν)ᶜ) = 0
· rw [hνs, mul_zero]
exact hμν hνs
· rw [ENNReal.top_mul hνs, measure_eq_top_of_subset_compl_sigmaFiniteSetWRT Set.inter_subset_right hνs]