Measure

In music:

A measure is a unit of time in Western music, also known as the bar. It represents a regular grouping of beats?, as indicated in notation by the time signature.

---

In mathematics:

A measure is a function from a class of subsets of a set to the real numbers in the range [0,inf). It is defined so as to give some idea of the relative "bigness" of the set. The intuitive idea of a measure is that little sets will have small measures, while bigger sets will have big measures.

A function is a measure iff:

• The value of the empty set is zero
• The sum of the measures of two disjoint sets is equal to the measure of the union of the two sets

Usually, mathematics deals with measures which also satisfy

• The sum of the measures of a countable sequence of pairwise-disjoint sets is the measure of their union

It is important to note that the axiom of choice implies that there is no measure on the set of all subsets of the real numbers which gives the right values to intervals.

There are several measures in common use:

• The Borel measure,
• The Lebesgue measure, which provides the foundation for modern theories of integration, and
• Probability measures (see Probability axioms)