English
For finite unions over Finset, the germ-indicator of the union converges to the germ-indicator of the infinite union, via atTop, mirroring the Monotone case.
Русский
Для конечного объединения по Finset индикатор Г germ сходится к индикатору бесконечного объединения, через atTop.
LaTeX
$$$(\\forall n: Finset ι)\\; (\\bigcup_{i∈n} s_i).mulIndicator f a$ tends to $(\\bigcup_i s_i).mulIndicator f a$$$
Lean4
@[to_additive]
theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
(fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (iUnion s) f a :=
by
rw [iUnion_eq_iUnion_finset s]
apply Monotone.mulIndicator_eventuallyEq_iUnion
exact fun _ _ ↦ biUnion_subset_biUnion_left