Definition:Space of Borel Probability Measures on Compact Metric Space

Definition

Let $\struct {X, d}$ be a compact metric space.

Let $\map \PP X$ be the set of all Borel probability measures on $X$.

Equip $\map \PP X$ with the weak-* topology, i.e. the initial topology with respect to:

$\ds \sequence { \mu \to \int f \rd \mu}_{f \in \map C {X,\R} }$

where $\map C {X,\R}$ denotes the space of real continuous functions.

Then $\map \PP X$ is called the space of Borel probability measures on $X$.

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $B.5:$ Measures on Compact Metric Spaces

Also see

• Space of Borel Probability Measures is Subspace of Dual of Continuous Functions: $\map \PP X \subseteq {\map C {X,\R} }^\ast$