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.

The inner measure induced by a measure

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:

  1. is non-negative,
  2. If then

Measure completion

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

See also


  1. ^ Halmos 1950, § 14, Theorem F