Measure (mathematics) - Examples

Examples

Some important measures are listed here.

  • The counting measure is defined by μ(S) = number of elements in S.
  • The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ = 1; and every other measure with these properties extends Lebesgue measure.
  • Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.
  • The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
  • The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
  • Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval ). Such a measure is called a probability measure. See probability axioms.
  • The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure and Young measure.

In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.

Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.

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