integrable — Mathlib

English

Density withDensity distributes over product of densities: κ.withDensity (f a x * g a x) equals (κ.withDensity f).withDensity g under measurability assumptions.

Русский

Плотность сDensity распределяется по произведению плотностей: κ.withDensity(f a x · g a x) = (κ.withDensity f).withDensity g при условиях измеримости.

LaTeX

$$$\\mathrm{withDensity}\\;\\kappa\\; (\\lambda a x. f(a,x) \\cdot g(a,x)) = \\mathrm{withDensity}\\bigl(\\mathrm{withDensity}\\kappa(\\lambda a x. f(a,x))\\bigr)(\\lambda a x. g(a,x))$$$

Lean4

protected theorem integrable (hf : Martingale f ℱ μ) (i : ι) : Integrable (f i) μ :=
  integrable_condExp.congr (hf.condExp_ae_eq (le_refl i))

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