measurable — Mathlib

English

If each function f_i is measurable and for every measurable set s we have IsLUB property on s, then the glued function is measurable on the complement as well.

Русский

Если каждый f_i измерим, и для каждого измеримого множества s выполняется свойство IsLUB на s, то соединённая функция измерима на дополнении.

LaTeX

$$$\forall i,\text{Measurable}(f_i) \Rightarrow \forall s,\ MeasurableSet(s) \Rightarrow \text{IsLUB}(g|_{s^c}) \Rightarrow \text{Measurable}(g|_{s^c})$$$

Lean4

theorem measurable [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α} (hf : LowerSemicontinuous f) :
    Measurable f :=
  measurable_of_Ioi fun y => (hf.isOpen_preimage y).measurableSet

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