Inner measure

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

φ :

defined on all subsets of a set

that satisfies the following conditions: Let

be a σ-algebra over a set

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

-measurable subsets.

Even though the set function

is usually not a measure,

shares the following properties with measures: Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra.

is a finite measure defined on a σ-algebra

are corresponding induced outer and inner measures, then the sets

form a σ-algebra

[1] The set function

defined by

is a measure on

known as the completion of