English
If s is closed and y ≤ 0, then the function indicator_s·y is lower semicontinuous.
Русский
Если s замкнуто и y ≤ 0, то функция индикатора s умноженная на y—нижнеполупреграммна (low semicontinuous).
LaTeX
$$$s\ ext{IsClosed},\; y \le 0 \Rightarrow \operatorname{LowerSemicontinuous}(\mathbf{1}_s \cdot y).$$$
Lean4
theorem lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) : LowerSemicontinuous (indicator s fun _x => y) :=
by
intro x z hz
by_cases h : x ∈ s <;> simp [h] at hz
· refine Filter.Eventually.of_forall fun x' => ?_
by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy]
· filter_upwards [hs.isOpen_compl.mem_nhds h]
simp +contextual [hz]