Inner measure

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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 ],} {\displaystyle \varphi :2^{X}\to [0,\infty ],} defined on all subsets of a set X , {\displaystyle 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} {\displaystyle \varphi (\varnothing )=0}
  • Superadditive: For any disjoint sets A {\displaystyle A} {\displaystyle A} and B , {\displaystyle B,} {\displaystyle B,} φ ( A ∪ B ) ≥ φ ( A ) + φ ( B ) . {\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (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 } {\displaystyle A_{1},A_{2},\ldots } of sets such that A j ⊇ A j + 1 {\displaystyle A_{j}\supseteq A_{j+1}} {\displaystyle A_{j}\supseteq A_{j+1}} for each j {\displaystyle j} {\displaystyle j} and φ ( A 1 ) < ∞ {\displaystyle \varphi (A_{1})<\infty } {\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})} {\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} {\displaystyle A} with φ ( A ) = ∞ {\displaystyle \varphi (A)=\infty } {\displaystyle \varphi (A)=\infty }, then this infinity must be approached. More precisely, if φ ( A ) = ∞ {\displaystyle \varphi (A)=\infty } {\displaystyle \varphi (A)=\infty } for a set A {\displaystyle A} {\displaystyle A} then for every positive real number r , {\displaystyle r,} {\displaystyle r,} there exists some B ⊆ A {\displaystyle B\subseteq A} {\displaystyle B\subseteq A} such that r ≤ φ ( B ) < ∞ . {\displaystyle r\leq \varphi (B)<\infty .} {\displaystyle r\leq \varphi (B)<\infty .}

The inner measure induced by a measure

Let Σ {\displaystyle \Sigma } {\displaystyle \Sigma } be a σ-algebra over a set X {\displaystyle X} {\displaystyle X} and μ {\displaystyle \mu } {\displaystyle \mu } be a measure on Σ . {\displaystyle \Sigma .} {\displaystyle \Sigma .} Then the inner measure μ ∗ {\displaystyle \mu _{*}} {\displaystyle \mu _{*}} induced by μ {\displaystyle \mu } {\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\}.} {\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}

Essentially μ ∗ {\displaystyle \mu _{*}} {\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 } {\displaystyle \mu }-measure of any of its Σ {\displaystyle \Sigma } {\displaystyle \Sigma }-measurable subsets. Even though the set function μ ∗ {\displaystyle \mu _{*}} {\displaystyle \mu _{*}} is usually not a measure, μ ∗ {\displaystyle \mu _{*}} {\displaystyle \mu _{*}} shares the following properties with measures:

  1. μ ∗ ( ∅ ) = 0 , {\displaystyle \mu _{*}(\varnothing )=0,} {\displaystyle \mu _{*}(\varnothing )=0,}
  2. μ ∗ {\displaystyle \mu _{*}} {\displaystyle \mu _{*}} is non-negative,
  3. If E ⊆ F {\displaystyle E\subseteq F} {\displaystyle E\subseteq F} then μ ∗ ( E ) ≤ μ ∗ ( F ) . {\displaystyle \mu _{*}(E)\leq \mu _{*}(F).} {\displaystyle \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 μ {\displaystyle \mu } {\displaystyle \mu } is a finite measure defined on a σ-algebra Σ {\displaystyle \Sigma } {\displaystyle \Sigma } over X {\displaystyle X} {\displaystyle X} and μ ∗ {\displaystyle \mu ^{*}} {\displaystyle \mu ^{*}} and μ ∗ {\displaystyle \mu _{*}} {\displaystyle \mu _{*}} are corresponding induced outer and inner measures, then the sets T ∈ 2 X {\displaystyle T\in 2^{X}} {\displaystyle T\in 2^{X}} such that μ ∗ ( T ) = μ ∗ ( T ) {\displaystyle \mu _{*}(T)=\mu ^{*}(T)} {\displaystyle \mu _{*}(T)=\mu ^{*}(T)} form a σ-algebra Σ ^ {\displaystyle {\hat {\Sigma }}} {\displaystyle {\hat {\Sigma }}} with Σ ⊆ Σ ^ {\displaystyle \Sigma \subseteq {\hat {\Sigma }}} {\displaystyle \Sigma \subseteq {\hat {\Sigma }}}.[1] The set function μ ^ {\displaystyle {\hat {\mu }}} {\displaystyle {\hat {\mu }}} defined by μ ^ ( T ) = μ ∗ ( T ) = μ ∗ ( T ) {\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)} {\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)} for all T ∈ Σ ^ {\displaystyle T\in {\hat {\Sigma }}} {\displaystyle T\in {\hat {\Sigma }}} is a measure on Σ ^ {\displaystyle {\hat {\Sigma }}} {\displaystyle {\hat {\Sigma }}} known as the completion of μ . {\displaystyle \mu .} {\displaystyle \mu .}

See also

  • Lebesgue measurable set – Broadest definition of sizes in integer-dimensional spacesPages displaying short descriptions of redirect targets

References

  1. Halmos 1950, § 14, Theorem F