inner — Mathlib

English

If f and g are strongly measurable, then the map t ↦ ⟪f(t), g(t)⟫ is strongly measurable.

Русский

Если функции f и g сильно измеримы, то карта t ↦ ⟪f(t), g(t)⟫ существенно измерима.

LaTeX

$$$\\text{StronglyMeasurable}(f) \\land \\text{StronglyMeasurable}(g) \\Rightarrow \\text{StronglyMeasurable}(t \\mapsto \\langle f(t), g(t) \\rangle)$$$

Lean4

@[fun_prop]
protected theorem inner {_ : MeasurableSpace α} {f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun t => ⟪f t, g t⟫ :=
  Continuous.comp_stronglyMeasurable continuous_inner (hf.prodMk hg)

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