mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an outer measure ''μ'' on ''n''-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

R^''n'' is called a Borel regular measure if the following two conditions hold: * Every

Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...

''B'' ⊆ R^''n'' is ''μ''-measurable in the sense of

Carathéodory's criterion Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable. Statement Carathéodory's criterion: Let \lambda^* : (\R^n) \to ...

: for every ''A'' ⊆ R^''n'', ::

\mu (A) = \mu (A \cap B) + \mu (A \setminus B).

* For every set ''A'' ⊆ R^''n'' there exists a Borel set ''B'' ⊆ R^''n'' such that ''A'' ⊆ ''B'' and ''μ''(''A'') = ''μ''(''B''). Notice that the set ''A'' need not be ''μ''-measurable: ''μ''(''A'') is however well defined as ''μ'' is an outer measure. An outer measure satisfying only the first of these two requirements is called a ''

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

'', while an outer measure satisfying only the second requirement (with the Borel set B replaced by a measurable set B) is called a ''

regular measure In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topol ...

''. The Lebesgue outer measure on R^''n'' is an example of a Borel regular measure. It can be proved that a Borel regular measure, although introduced here as an ''outer'' measure (only countably ''sub''additive), becomes a full measure ( countably additive) if restricted to the

References

* * * {{Measure theory Measures (measure theory)