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. Definition. An inner measure is a set function formula_1 defined on all subsets of a set formula_2 that satisfies the following conditions: The inner measure induced by a measure. Let formula_17 be a σ-algebra over a set formula_18 and formula_19 be a measure on formula_20 Then the inner  measure formula_21 induced by formula_19 is defined by formula_23 Essentially formula_21 gives a lower bound of the size of any set by ensuring it is at least as big as the formula_19-measure of any of its formula_17-measurable subsets. Even though the set function formula_21 is usually not a measure, formula_21 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 formula_19 is a finite measure defined on a σ-algebra formula_17 over formula_18 and formula_36 and formula_21 are corresponding  induced outer and inner measures, then the sets formula_38 such that formula_39 form a σ-algebra formula_40 with formula_41. The set function formula_42 defined by formula_43 for all formula_44 is a measure on formula_40 known as the completion of formula_46