Measure Theory/Measures on topological spaces

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Definition (Borel σ-algebra):

Let be a topological space. The Borel -algebra on is the -algebra generated by all open subsets of , ie.

,

where is the topology on .

Definition (tight):

Let be a topological space and let be a -algebra on that contains the Borel -algebra. A measure is called tight iff for all sets

.

The following proposition provides a class of tight measure spaces:

Proposition (Borel measure on Polish space is tight):

Let Failed to parse (syntax error): {\displaystyle {{definition|inner regular|Let <math>\Omega} be a topological space and let be a -algebra on that contains the Borel -algebra. A measure is called inner regular iff for all sets

.

Definition (outer regular):

Let be a topological space and let be a -algebra on that contains the Borel -algebra. A measure is called outer regular iff for all sets

.

Proposition (closed set with empty interior in σ-compact measure space is nullset):

Let be a topological space, let be a -algebra on that contains the Borel -algebra, and suppose that is a ... measure on . Then every closed subset that has empty interior is a nullset.

Proof: Let

,

where the are compact. Then we have by countable subadditivity of measure

.

But closed subsets of compact sets are compact, and hence it suffices to prove that whenever is a closed, compact subset of .

Exercises[edit]