In mathematics, an outer measure *μ* on *n*-dimensional Euclidean space **R**^{n} is called a **Borel regular measure** if the following two conditions hold:

- Every Borel set
*B*⊆**R**^{n}is*μ*-measurable in the sense of Carathéodory's criterion: for every*A*⊆**R**^{n},

- 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*, 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*.

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 Borel sets.