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 $$\varphi : 2^X \to [0, \infty],$$ defined on all subsets of a set $$X,$$ that satisfies the following conditions:


 * Null empty set: The empty set has zero inner measure (see also: measure zero); that is, $$\varphi(\varnothing) = 0$$
 * Superadditive: For any disjoint sets $$A$$ and $$B,$$ $$\varphi(A \cup B) \geq \varphi(A) + \varphi(B).$$
 * Limits of decreasing towers: For any sequence $$A_1, A_2, \ldots$$ of sets such that $$ A_j \supseteq A_{j+1}$$ for each $$j$$ and $$\varphi(A_1) < \infty$$ $$\varphi \left(\bigcap_{j=1}^\infty A_j\right) = \lim_{j \to \infty} \varphi(A_j)$$
 * If the measure is not finite, that is, if there exist sets $$A$$ with $$\varphi(A) = \infty$$, then this infinity must be approached. More precisely, if $$\varphi(A) = \infty$$ for a set $$A$$ then for every positive real number $$r,$$ there exists some $$B \subseteq A$$ such that $$r \leq \varphi(B) < \infty.$$

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 $$\mu_*(T) = \sup\{\mu(S) : S \in \Sigma \text{ and } S \subseteq T\}.$$

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:
 * 1) $$\mu_*(\varnothing) = 0,$$
 * 2) $$\mu_*$$ is non-negative,
 * 3) If $$E \subseteq F$$ then $$\mu_*(E) \leq \mu_*(F).$$

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 $$\mu_*(T) = \mu^*(T)$$ form a σ-algebra \hat \Sigma with. The set function $$\hat\mu$$ defined by $$\hat\mu(T) = \mu^*(T) = \mu_*(T)$$ for all is a measure on \hat \Sigma known as the completion of $$\mu.$$