English
For Polish spaces, a finite measure μ is inner regular with respect to compact and closed sets: for every measurable A, μ(A) = sup { μ(K) : K ⊆ A, K is compact and closed }.
Русский
Для поля пространства, где полская топология применяется, конечная мера μ является внутренне регулярной по компактно-замкнутым множествам: для каждого измеримого множества A имеет место μ(A) = sup { μ(K) : K ⊆ A, K компактно и замкнуто }.
LaTeX
$$$\forall A \in \mathcal{M}(\alpha),\; μ(A) = \sup\{ μ(K) : K \subseteq A,\ K \text{ compact and closed} \}.$$$
Lean4
/-- A special case of `innerRegularCompactLTTop_of_pseudoEMetricSpace_completeSpace_secondCountable`
for Polish spaces: A measure `μ` on a Polish space inner regular for finite measure sets with
respect to compact sets.
-/
instance InnerRegularCompactLTTop_of_polishSpace [TopologicalSpace α] [PolishSpace α] [BorelSpace α] (μ : Measure α) :
μ.InnerRegularCompactLTTop :=
by
letI := TopologicalSpace.upgradeIsCompletelyMetrizable α
exact InnerRegularCompactLTTop_of_pseudoEMetricSpace_completeSpace_secondCountable μ
