Measure (Mathematics)

In mathematics, a measure is a way of assigning a positive real number (or 0) to sets of numbers. In order to be called a measure, such a way of assigning numbers to sets (called a "function") must satisfy the conditions:

• the measure of an empty set (a set with no members) is 0.
• if two sets do not have any members in common (i.e., are disjoint), then measure of both sets together is just the sum of the measures of each of those sets individually.

Lebesgue Measure

The most common measure of sets of real numbers is Lebesgue measure, which was deliberately constructed as a measure which would measure intervals as their length and points as 0 - it is the most "intuitional" measure of real numbers.

Measurability

If we accept the axiom of choice, it is possible to construct sets of numbers for which no function can satisfy the second condition. Thus, the domain of a measure is not all sets of real numbers, but only some of them. For the most common measure on real numbers, the Lebesgue measure, these sets are simply called Lebesgue measurable.