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
φ : 2 X → [ 0 , ∞ ] , {\displaystyle \varphi :2^{X}\to [0,\infty ],}
defined on all subsets of a set X , {\displaystyle X,} that satisfies the following conditions:
Null empty set: The empty set has zero inner measure (see also: measure zero); that is, φ ( ∅ ) = 0 {\displaystyle \varphi (\varnothing )=0}
Superadditive: For any disjoint sets A {\displaystyle A} and B , {\displaystyle B,} φ ( A ∪ B ) ≥ φ ( A ) + φ ( B ) . {\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (B).}
Limits of decreasing towers: For any sequence A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } of sets such that A j ⊇ A j + 1 {\displaystyle A_{j}\supseteq A_{j+1}} for each j {\displaystyle j} and φ ( A 1 ) < ∞ {\displaystyle \varphi (A_{1})<\infty } φ ( ⋂ j = 1 ∞ A j ) = lim j → ∞ φ ( A j ) {\displaystyle \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 {\displaystyle A} with φ ( A ) = ∞ {\displaystyle \varphi (A)=\infty } , then this infinity must be approached. More precisely, if φ ( A ) = ∞ {\displaystyle \varphi (A)=\infty } for a set A {\displaystyle A} then for every positive real number r , {\displaystyle r,} there exists some B ⊆ A {\displaystyle B\subseteq A} such that r ≤ φ ( B ) < ∞ . {\displaystyle r\leq \varphi (B)<\infty .}
The inner measure induced by a measure Let Σ {\displaystyle \Sigma } be a σ-algebra over a set X {\displaystyle X} and μ {\displaystyle \mu } be a measure on Σ . {\displaystyle \Sigma .} Then the inner measure μ ∗ {\displaystyle \mu _{*}} induced by μ {\displaystyle \mu } is defined by
μ ∗ ( T ) = sup { μ ( S ) : S ∈ Σ and S ⊆ T } . {\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}
Essentially μ ∗ {\displaystyle \mu _{*}} gives a lower bound of the size of any set by ensuring it is at least as big as the μ {\displaystyle \mu } -measure of any of its Σ {\displaystyle \Sigma } -measurable subsets. Even though the set function μ ∗ {\displaystyle \mu _{*}} is usually not a measure, μ ∗ {\displaystyle \mu _{*}} shares the following properties with measures:
μ ∗ ( ∅ ) = 0 , {\displaystyle \mu _{*}(\varnothing )=0,}
μ ∗ {\displaystyle \mu _{*}} is non-negative, If E ⊆ F {\displaystyle E\subseteq F} then μ ∗ ( E ) ≤ μ ∗ ( F ) . {\displaystyle \mu _{*}(E)\leq \mu _{*}(F).}
Measure completion
… excerpt ends here. Continue reading the full article.
