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.

Definition

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

The inner measure induced by a measure

Let \Sigma be a σ-algebra over a set X and \mu be a measure on \Sigma. Then the inner measure \mu_* induced by \mu is defined by Essentially \mu_* gives a lower bound of the size of any set by ensuring it is at least as big as the \mu-measure of any of its \Sigma-measurable subsets. Even though the set function \mu_* is usually not a measure, \mu_* shares the following properties with measures:

Measure completion

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If \mu is a finite measure defined on a σ-algebra \Sigma over X and \mu^* and \mu_* are corresponding induced outer and inner measures, then the sets T \in 2^X such that form a σ-algebra \hat \Sigma with. The set function \hat\mu defined by for all is a measure on \hat \Sigma known as the completion of \mu.