This makes the regularity conditions unnecessary as Baire measures are automatically regular.
2.
In particular, any compactly supported continuous function on such a space is integrable with respect to any Baire measure.
3.
There are several inequivalent definitions of Baire sets, so correspondingly there are several inequivalent concepts of Baire measure on a topological space.
4.
Thus, measures defined on this ?-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces.
5.
In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.
6.
For every compact Hausdorff space, every finite Baire measure ( that is, a measure on the ?-algebra of all Baire sets ) is regular.
7.
One reason for working with compact " G " ? sets rather than closed " G " ? sets is that Baire measures are then automatically regular.
8.
Other'named'measures used in various theories include : Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and strong measure zero.
9.
If " X " is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued functions on " X ", by the Riesz Markov Kakutani representation theorem.
10.
However, in this case it is no longer true that a finite Baire measure is necessarily regular : for example the Baire probability measure that assigns measure 0 to every countable subset of an uncountable discrete space and measure 1 to every co-countable subset is a Baire probability measure that is not regular.
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