map_fst_prod — Mathlib

English

The product of a finite measure with the zero finite measure is zero: μ.prod (0 : FiniteMeasure β) = 0.

Русский

Произведение конечной меры с нулевой конечной мерой равно нулю: μ.prod (0 : FiniteMeasure β) = 0.

LaTeX

$$$\mu \mathrm{prod} (0 : \text{FiniteMeasure } \beta) = 0$$$

Lean4

/-- The first marginal of a product probability measure is the first probability measure. -/
@[simp]
theorem map_fst_prod : (μ.prod ν).map measurable_fst.aemeasurable = μ :=
  by
  apply Subtype.ext
  simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_fst_prod, measure_univ, one_smul]

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