pi_Ico_ae_eq_pi_Icc — Mathlib

English

A mixed product over an Option-indexed family is equal to the product over the Option-indexed family after applying the canonical equivalence piOptionEquivProd.

Русский

Смешанное произведение над семейством с индексом Option равно произведению после применения канонического эквивалента piOptionEquivProd.

LaTeX

$$$((\\mathrm{pi}\\; (\\lambda i.\\; \\mu(\\mathrm{some}\\ i))).\\mathrm{prod}} (\\mu\\; \\mathrm{none})) \\,\\map (\\mathrm{MeasurableEquiv.piOptionEquivProd}\\; \\beta).\\mathrm{symm} = \\mathrm{Measure.pi}\\; \\mu$$$

Lean4

theorem pi_Ico_ae_eq_pi_Icc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ico (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Icc (f i) (g i) :=
  ae_eq_set_pi fun _ _ => Ico_ae_eq_Icc

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