Baire measure
Encyclopedia
A Baire measure is a measure
on the σ-algebra of Baire set
s of a topological space
. In spaces that are not metric space
s, the Borel set
s and the Baire set
s may differ. In this context, Baire measures are used because they connect to the properties of continuous function
s more directly.
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
on the σ-algebra of Baire set
Baire set
In mathematics, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory and topology. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological...
s of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. In spaces that are not metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s, the Borel set
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
s and the Baire set
Baire set
In mathematics, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory and topology. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological...
s may differ. In this context, Baire measures are used because they connect to the properties of continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s more directly.