In mathematics, in particular in measure theory, an **inner measure** is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.

An inner measure is a set function defined on all subsets of a set that satisfies the following conditions:

- Null empty set: The empty set has zero inner measure (
*see also: measure zero*); that is, - Superadditive: For any disjoint sets and
- Limits of decreasing towers: For any sequence of sets such that for each and
- If the measure is not finite, that is, if there exist sets with , then this infinity must be approached. More precisely, if for a set then for every positive real number there exists some such that

Let be a σ-algebra over a set and be a measure on Then the inner measure induced by is defined by

Essentially gives a lower bound of the size of any set by ensuring it is at least as big as the -measure of any of its -measurable subsets. Even though the set function is usually not a measure, shares the following properties with measures:

- is non-negative,
- If then

Main article: Complete measure |

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If is a finite measure defined on a σ-algebra over and and are corresponding induced outer and inner measures, then the sets such that form a σ-algebra with .^{[1]}
The set function defined by
for all is a measure on known as the completion of