lowerSemicontinuousAt_iSup — Mathlib

English

If every f_i is lower semicontinuous at x, then the supremum over i of f_i is lower semicontinuous at x.

Русский

Если каждая f_i лекс. полупрерывна в x, то их надмножество верхов supremum тоже лс в x.

LaTeX

$$$\\displaystyle \\Big(\\forall i,\\ LowerSemicontinuousAt(f_i,x)\\Big) \\Rightarrow \\ LowerSemicontinuousAt\\Big( x' \\mapsto \\bigsqcup_i f_i(x') , x \\Big)$$$

Lean4

theorem lowerSemicontinuousAt_iSup {f : ι → α → δ} (h : ∀ i, LowerSemicontinuousAt (f i) x) :
    LowerSemicontinuousAt (fun x' => ⨆ i, f i x') x :=
  lowerSemicontinuousAt_ciSup (by simp) h

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