pi_map_pi — Mathlib

English

Similarly for Real-valued product measures, the pi construction yields a real-valued product.

Русский

Аналогично для вещественных произведений мер, конструктор пи дает вещественное произведение.

LaTeX

$$toMeasure_pi: (ProbabilityMeasure.pi μ).toMeasure = Measure.pi (μ_i.toMeasure)$$

Lean4

theorem pi_map_pi {β : ι → Type*} [∀ i, MeasurableSpace (β i)] {f : Π i, α i → β i}
    (f_mble : ∀ i, AEMeasurable (f i) (μ i)) :
    (FiniteMeasure.pi μ).map (fun x i ↦ (f i (x i))) = FiniteMeasure.pi (fun i ↦ (μ i).map (f i)) :=
  by
  apply Subtype.ext
  simp only [val_eq_toMeasure, toMeasure_map, toMeasure_pi]
  rw [Measure.pi_map_pi f_mble]

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