add_prod — Mathlib

English

The product with the zero measure in the second factor is zero: μ.prod 0 = 0.

Русский

Произведение с нулевой мерой во втором факторе равно нулю: μ.prod 0 = 0.

LaTeX

$$$μ.prod 0 = 0$$$

Lean4

theorem add_prod (μ' : Measure α) [SFinite μ'] : (μ + μ').prod ν = μ.prod ν + μ'.prod ν :=
  by
  simp_rw [← sum_sfiniteSeq μ, ← sum_sfiniteSeq μ', sum_add_sum, ← sum_sfiniteSeq ν, prod_sum, sum_add_sum]
  congr
  ext1 i
  refine prod_eq fun s t _ _ => ?_
  simp_rw [add_apply, prod_prod, right_distrib]

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