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In mathematics, a measure is a function that assigns "sizes", "volumes", or "probabilities" to subsets of a given set. The concept is important in mathematical analysis and probability theory.

Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and integrals. It is of importance in probability and statistics.

## Formal definitions

Formally, a measure μ is a function which assigns to every element S of a given sigma algebra X a value μ(S), a non-negative real number or ∞. The following properties have to be satisfied:

• The empty set has measure zero: μ({}) = 0.
• The measure is countably additive: if E1, E2, E3, ... are countably many pairwise disjoint sets in X and E is their union, then the measure μ(E) is equal to the sum ∑μ(Ek).

If μ is a measure on the sigma algebra X, then the members of X are called the μ-measurable sets, or the measurable sets for short. A set Ω together with a sigma algebra X on Ω and a measure μ on X is called a measure space.

The following properties can be derived from the definition above:

• If E1 and E2 are two measurable sets with E1 being a subset of E2, then μ(E1) ≤ μ(E2).
• If E1, E2, E3, ... are measurable sets and En is a subset of En+1 for all n, then the union E of the sets En is measurable and μ(E) = lim μ(En).
• If E1, E2, E3, ... are measurable sets and En+1 is a subset of En for all n, then the intersection E of the sets En is measurable; furthermore, if at least one of the En has finite measure, then μ(E) = lim μ(En).

A measure space Ω is called finite if μ(Ω) is a finite real number (rather than ∞). It is called σ-finite if Ω is the countable union of measurable sets of finite measure.

σ-finite measure spaces have some very nice properties; σ-finiteness can be compared in this respect to separable of topological spaces.

A measurable set S is called a null-set if μ(S) = 0. The measure μ is called complete if every subset of a null-set is measurable (and then automatically itself a null-set).

### Examples

Some important measures are listed here.

### Generalizations

For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. A measure that takes values in a Banach space is called a spectral measure; these are used mainly in functional analysis for the spectral theorem.

Another generalization is the finitely additive measure. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful.

The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in Rn consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k=0,1,2,...,n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c>0 multiplies the set's "measure" by ck. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n-1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogenous of degree 0 is the Euler characteristic.