- Definition. Let
be a nonempty set. An outer measure [1] on the set
is a function
such that
,
if
,

The second and third conditions in the definition of an outer measure are equivalent to the condition that
implies
.
- Definition. A set
is called
-measurable if
for all
.
Examples of Outer Measures
The standard example of an outer measure is the Lebesgue outer measure, defined on subsets of
.

A near-generalization of the Lebesgue outer measure is given by
![{\displaystyle \mu _{F}^{*}(A)=\inf \left\{\sum _{i=1}^{\infty }|F(b_{i})-F(a_{i})|:A\subseteq \bigcup _{i=1}^{\infty }(a_{i},b_{i}]\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f663363eb887ffffd6bb2181fdde85032c8c8687)
where
is any right-continuous function [2].
Given a measure space
, one can always define an outer measure
[3] by

References
- ↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.4
- ↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.5
- ↑ Craig, Katy. MATH 201A HW 3. UC Santa Barbara, Fall 2020.