English
The outer measure of t under the PMF from s equals the sum over all x of the count of x in s restricted to t, divided by card(s).
Русский
Внешняя мера множества t по PMF из s равна сумме количеств элементов s, попадающих в t, деленной на card(s).
LaTeX
$$$$ (\\mathrm{toOuterMeasure\\_ofMultiset\\_apply}\\; s\\; hs).toOuterMeasure\\; t = \\frac{\\mathrm{tsum}\\; x\\; (s.\\mathrm{count}\\; x\\; (\\mathrm{Multiset}.filter (\\lambda x. x \\in t) s))}{(\\mathrm{Multiset}.card\\; s)} $$$$
Lean4
@[to_additive]
theorem hasLaw_mul {M : Type*} [Monoid M] {mM : MeasurableSpace M} [MeasurableMul₂ M] {μ ν : Measure M} [SigmaFinite μ]
[SigmaFinite ν] {X Y : Ω → M} (hX : HasLaw X μ P) (hY : HasLaw Y ν P) (hXY : IndepFun X Y P) :
HasLaw (X * Y) (μ ∗ₘ ν) P where
map_eq := by
rw [hXY.map_mul_eq_map_mconv_map₀' hX.aemeasurable hY.aemeasurable, hX.map_eq, hY.map_eq]
· rwa [hX.map_eq]
· rwa [hY.map_eq]
