prod_dirac — Mathlib

English

The product of two dirac measures equals the dirac measure at the pair: dirac x × dirac y = dirac (x, y).

Русский

Произведение двух дирек мер равно дирек мере по паре: dirac x × dirac y = dirac (x, y).

LaTeX

$$$(\\mathrm{dirac}\\,x) \\prod (\\mathrm{dirac}\\,y) = \\mathrm{dirac}(x,y)$$$

Lean4

theorem prod_dirac (y : β) : μ.prod (dirac y) = map (fun x => (x, y)) μ := by
  classical
  rw [← sum_sfiniteSeq μ, prod_sum_left, map_sum measurable_prodMk_right.aemeasurable]
  congr
  ext1 i
  refine prod_eq fun s t hs ht => ?_
  simp_rw [map_apply measurable_prodMk_right (hs.prod ht), mk_preimage_prod_left_eq_if, measure_if, dirac_apply' _ ht, ←
    indicator_mul_right _ fun _ => sfiniteSeq μ i s, Pi.one_apply, mul_one]

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