nullMeasurableSet_prod_of_ne_zero

English

The product of three measures is associative up to the natural isomorphism: map prodAssoc ((μ.prod ν).prod τ) equals μ.prod (ν.prod τ).

Русский

Произведения трех мер ассоциативны при естественном изоморфизме: отображение prodAssoc переводит ((μ × ν) × τ) во μ × (ν × τ).

LaTeX

$$$\\operatorname{map}(\\text{MeasurableEquiv.prodAssoc})\\big((μ \\prod ν) \\prod τ\\big) = μ \\prod (ν \\prod τ)$$$

Lean4

/-- The product of two non-null sets is null measurable
if and only if both of them are null measurable. -/
theorem nullMeasurableSet_prod_of_ne_zero {s : Set α} {t : Set β} (hs : μ s ≠ 0) (ht : ν t ≠ 0) :
    NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν :=
  ⟨fun h ↦ ⟨h.left_of_prod ht, h.right_of_prod hs⟩, fun ⟨hs, ht⟩ ↦ hs.prod ht⟩

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