condExp_natural_ae_eq_of_lt — Mathlib

English

If a sequence of independent sets (s_n) has infinite sum of measures, then limsup s_n has measure one.

Русский

Если последовательность независимых множеств (s_n) имеет бесконечную сумму мер, то limsup s_n имеет меру зеро 1.

LaTeX

$$$$\text{Theorem measure_limsup_eq_one: } \text{Given } (s_n) \text{ independent with } \sum \mu(s_n)=\infty,\; \mu(\limsup s_n)=1.$$$

Lean4

theorem condExp_natural_ae_eq_of_lt [SecondCountableTopology β] [CompleteSpace β] [NormedSpace ℝ β]
    (hf : ∀ i, StronglyMeasurable (f i)) (hfi : iIndepFun f μ) (hij : i < j) :
    μ[f j|Filtration.natural f hf i] =ᵐ[μ] fun _ => μ[f j] :=
  by
  have : IsProbabilityMeasure μ := hfi.isProbabilityMeasure
  exact
    condExp_indep_eq (hf j).measurable.comap_le (Filtration.le _ _) (comap_measurable <| f j).stronglyMeasurable
      (hfi.indep_comap_natural_of_lt hf hij)

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