English
Let {μ_i} i∈I be a countable family of outer measures. Then trimming commutes with supremum: the trim of the supremum equals the supremum of the trims.
Русский
Пусть {μ_i} индексируемая счётно выходящаяся семейство внешних мер. Тогда обрезка сохраняет супрему: обрезка сверху равна супрему обрезок.
LaTeX
$$$\\operatorname{trim}\\left(\\big\\l-sup_{i} \\mu_i\\big) = \\big\\supsup_{i} \\operatorname{trim}(\\mu_i).$$$
Lean4
/-- `trim` sends the supremum of a countable family of outer measures to the supremum
of the trimmed measures. -/
theorem trim_iSup {ι} [Countable ι] (μ : ι → OuterMeasure α) : trim (⨆ i, μ i) = ⨆ i, trim (μ i) :=
by
simp_rw [← @iSup_plift_down _ ι]
ext1 s
obtain ⟨t, _, _, hμt⟩ :=
exists_measurable_superset_forall_eq_trim (Option.elim' (⨆ i, μ (PLift.down i)) (μ ∘ PLift.down)) s
simp only [Option.forall, Option.elim'] at hμt
simp only [iSup_apply, ← hμt.1]
exact iSup_congr hμt.2
