ext_of_forall_mem_subalgebra_integral_eq_of_polish

English

Extends the Polish-space-based ext lemma to the pseudo-metric complete countable setting.

Русский

Расширение леммы на полиш-пространство до полного псевдометрического счётного случая.

LaTeX

$$ext_of_forall_mem_subalgebra_integral_eq_of_polish ...$$

Lean4

theorem ext_of_forall_mem_subalgebra_integral_eq_of_polish [TopologicalSpace E] [PolishSpace E] [BorelSpace E]
    {P P' : Measure E} [IsFiniteMeasure P] [IsFiniteMeasure P'] {A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)}
    (hA : (A.map (toContinuousMapStarₐ 𝕜)).SeparatesPoints)
    (heq : ∀ g ∈ A, ∫ x, (g : E → 𝕜) x ∂P = ∫ x, (g : E → 𝕜) x ∂P') : P = P' :=
  by
  letI := TopologicalSpace.upgradeIsCompletelyMetrizable E
  exact ext_of_forall_mem_subalgebra_integral_eq_of_pseudoEMetric_complete_countable hA heq

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