In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
Definitions and first consequences Given a field of sets ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} and a Banach space X , {\displaystyle X,} a finitely additive vector measure (or measure, for short) is a function μ : F → X {\displaystyle \mu :{\mathcal {F}}\to X} such that for any two disjoint sets A {\displaystyle A} and B {\displaystyle B} in F {\displaystyle {\mathcal {F}}} one has
μ ( A ∪ B ) = μ ( A ) + μ ( B ) . {\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}
A vector measure μ {\displaystyle \mu } is called countably additive if for any sequence ( A i ) i = 1 ∞ {\displaystyle (A_{i})_{i=1}^{\infty }} of disjoint sets in F {\displaystyle {\mathcal {F}}} such that their union is in F {\displaystyle {\mathcal {F}}} it holds that
μ ( ⋃ i = 1 ∞ A i ) = ∑ i = 1 ∞ μ ( A i ) {\displaystyle \mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})}
with the series on the right-hand side convergent in the norm of the Banach space X . {\displaystyle X.}
It can be proved that an additive vector measure μ {\displaystyle \mu } is countably additive if and only if for any sequence ( A i ) i = 1 ∞ {\displaystyle (A_{i})_{i=1}^{\infty }} as above one has
where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the norm on X . {\displaystyle X.}
Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval [ 0 , ∞ ) , {\displaystyle [0,\infty ),} the set of real numbers, and the set of complex numbers.
Examples Consider the field of sets made up of the interval [ 0 , 1 ] {\displaystyle [0,1]} together with the family F {\displaystyle {\mathcal {F}}} of all Lebesgue measurable sets contained in this interval. For any such set A , {\displaystyle A,} define
μ ( A ) = χ A {\displaystyle \mu (A)=\chi _{A}}
where χ A {\displaystyle \chi _{A}} is the indicator function of A . {\displaystyle A.} Depending on where μ {\displaystyle \mu } is declared to take values, two different outcomes are observed.
μ , {\displaystyle \mu ,} viewed as a function from F {\displaystyle {\mathcal {F}}} to the L p {\displaystyle L^{p}} -space L ∞ ( [ 0 , 1 ] ) , {\displaystyle L^{\infty }([0,1]),} is a vector measure which is not countably-additive.
μ , {\displaystyle \mu ,} viewed as a function from F {\displaystyle {\mathcal {F}}} to the L p {\displaystyle L^{p}} -space L 1 ( [ 0 , 1 ] ) , {\displaystyle L^{1}([0,1]),} is a countably-additive vector measure. Both of these statements follow quite easily from the criterion (*) stated above.
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