# Measure Theory/Measures on topological spaces

**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 .