pi_Ioo_ae_eq_pi_Ioc — Mathlib

English

The a.e. equality transfers from Ioc to Icc in the product space: the product of Ioc(f_i) across I is a.e. equal to the product of Icc(f_i).

Русский

ae-равенство переносится из Ioc к Icc в произведении: произведение Ioc(f_i) равно по a.e. произведению Icc(f_i).

LaTeX

$$$(\\mathrm{pi}\\; s\\; (\\lambda i.\\; \\mathrm{Ioc}(f_i))) =_{a.e.}^{\\mathrm{Measure.pi}\\; \\mu} \\mathrm{pi}\\; s\\; (\\lambda i.\\; \\mathrm{Icc}(f_i))$$$

Lean4

theorem pi_Ioo_ae_eq_pi_Ioc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ioo (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Ioc (f i) (g i) :=
  ae_eq_set_pi fun _ _ => Ioo_ae_eq_Ioc

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