Baire measure
In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly. Variations. There are several inequivalent definitions of  Baire sets, so correspondingly there are several inequivalent concepts of Baire measure on a topological space. These all coincide on spaces that are locally compact σ-compact Hausdorff spaces. Relation to Borel measure. In practice Baire measures can be replaced by regular Borel measures. The relation between Baire measures and regular Borel measures is as follows: