Editing Measure (mathematics) (section)

==Instances==
{{main category|Measures (measure theory)}}
Some important measures are listed here.

* The [[counting measure]] is defined by <math>\mu(S)</math> = number of elements in <math>S.</math>
* The [[Lebesgue measure]] on <math>\R</math> is a [[Complete measure|complete]] [[translational invariance|translation-invariant]] measure on a ''σ''-algebra containing the [[interval (mathematics)|interval]]s in <math>\R</math> such that <math>\mu([0, 1]) = 1</math>; and every other measure with these properties extends the 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 space|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.
*Every (pseudo) [[Riemannian manifold]] <math>(M,g)</math> has a canonical measure <math>\mu_g</math> that in local coordinates <math>x_1,\ldots,x_n</math> looks like <math>\sqrt{\left|\det g \right|}d^nx</math> where <math>d^nx</math> is the usual Lebesgue measure.
* 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]] [0, 1]). Such a measure is called a ''probability measure'' or ''distribution''. See the [[list of probability distributions]] for instances.
* The [[Dirac measure]] ''δ''<sub>''a''</sub> (cf. [[Dirac delta function]]) is given by ''δ''<sub>''a''</sub>(''S'') = ''χ''<sub>''S''</sub>(a), where ''χ''<sub>''S''</sub> is the [[indicator function]] of <math>S.</math> The measure of a set is 1 if it contains the point <math>a</math> and 0 otherwise.

Other 'named' measures used in various theories include: [[Borel measure]], [[Jordan measure]], [[ergodic measure]], [[Gaussian measure]], [[Baire measure]], [[Radon measure]], [[Young measure]], and [[Loeb measure]].
 
In physics an example of a measure is spatial distribution of [[mass]] (see for example, [[gravity potential]]), or another non-negative [[extensive property]], [[conserved quantity|conserved]] (see [[Conservation law (physics)|conservation law]] for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

* [[Liouville's theorem (Hamiltonian)#Symplectic geometry|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]].

Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet.<ref>{{cite arXiv | eprint=2111.09266 | last1=Bengio | first1=Yoshua | last2=Lahlou | first2=Salem | last3=Deleu | first3=Tristan | last4=Hu | first4=Edward J. | last5=Tiwari | first5=Mo | last6=Bengio | first6=Emmanuel | title=GFlowNet Foundations | date=2021 | class=cs.LG }}</ref>