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