Measure Theory/Measures on topological spaces

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 . 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: Borel measure on Polish space is tight Let 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 . 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 . 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. 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