English
The triple product under a rearrangement is preserved by the prodAssoc isomorphism on measures.
Русский
Перестановка факторов в троечном произведении сохраняется через изоморфизм prodAssoc на мерах.
LaTeX
$$$\\text{ProdAssoc}$ induces equality: $\\text{map}(\\text{prodAssoc})((μ.prod ν).prod τ) = μ.prod (ν.prod τ)$$$
Lean4
/-- The product of two sets is null measurable
if and only if both of them are null measurable or one of them has measure zero. -/
theorem nullMeasurableSet_prod {s : Set α} {t : Set β} :
NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0 :=
by
rcases eq_or_ne (μ s) 0 with hs | hs; · simp [NullMeasurableSet.of_null, *]
rcases eq_or_ne (ν t) 0 with ht | ht; · simp [NullMeasurableSet.of_null, *]
simp [*, nullMeasurableSet_prod_of_ne_zero]
