24.7. VECTOR MEASURES 679

is said to be a vector measure if

F (∪∞i=1Ei) =

∞

∑i=1

F (Ei)

whenever {Ei}∞

i=1 is a sequence of disjoint elements of S . For F a vector measure,

|F |(A)≡ sup{ ∑F∈π(A)

∥µ (F)∥ : π (A) is a partition of A}.

This is the same definition that was given in the case where F would have values in C,the only difference being the fact that now F has values in a general Banach space X asthe vector space of values of the vector measure. Recall that a partition of A is a finite set,{F1, · · · ,Fm} ⊆S such that ∪m

i=1Fi = A. The same theorem about |F | proved in the case ofcomplex valued measures holds in this context with the same proof. For completeness, it isincluded here.

Theorem 24.7.2 If |F |(Ω)< ∞, then |F | is a measure on S .

Proof: Let E1 and E2 be sets of S such that E1∩E2 = /0 and let {Ai1, · · · ,Ai

ni}= π(Ei),

a partition of Ei which is chosen such that

|F |(Ei)− ε <ni

∑j=1∥F(Ai

j)∥ i = 1,2.

Consider the sets which are contained in either of π (E1) or π (E2) , it follows this collectionof sets is a partition of E1 ∪E2 which is denoted here by π(E1 ∪E2). Then by the aboveinequality and the definition of total variation,

|F |(E1∪E2)≥ ∑F∈π(E1∪E2)

∥F(F)∥> |F |(E1)+ |F |(E2)−2ε ,

which shows that since ε > 0 was arbitrary,

|F |(E1∪E2)≥ |F |(E1)+ |F |(E2). (24.30)

Let{

E j}∞

j=1 be a sequence of disjoint sets of S and let E∞ = ∪∞j=1E j. Then by the

definition of total variation there exists a partition of E∞, π(E∞) = {A1, · · · ,An} such that

|F |(E∞)− ε <n

∑i=1∥F(Ai)∥ .

Also,

Ai = ∪∞j=1Ai∩E j, so F (A j) =

∞

∑j=1

F (Ai∩E j)

and so by the triangle inequality, ∥F(Ai)∥ ≤ ∑∞j=1∥∥F(Ai∩E j)

∥∥. Therefore, by the above,

|F |(E∞)− ε <n

∑i=1

≥∥F(Ai)∥︷ ︸︸ ︷∞

∑j=1

∥∥F(Ai∩E j)∥∥= ∞

∑j=1

n

∑i=1

∥∥F(Ai∩E j)∥∥≤ ∞

∑j=1|F |(E j)