On the Sobczyk-Hammer Decomposition of Additive Set Functions

Abstract

Abstract. It is observed that continuity for charges is equivalent to the absence of two-valued minorants. This characterization forms the basis of a new short proof within a functional-analytic context of a decomposition theorem by A. Sobczyk and P. C. Hammer [5] for charges on a field ÍI into a continuous part and a part which can be written as a sum of at most two-valued charges on 51. A counterexample shows that in general the decomposition of a charge into a nonatomic part and a part which has no nonnull nonatomic minorant is not unique.

Note on Continuous and Purely Finitely Additive Set Functions

“…(2) Whereas nonatomicity is preserved with respect to limits of sequences of measures under the topology of setwise convergence (notice that for nonatomic measures | JL" and a measure |x 0 on a cr-field < § such that [x"(A) -» JJL 0 (A ) for any A E <S the measure |x = 2^=, 1/2" |x n is also nonatomic and |x 0 is absolutely continuous with respect to |x -then apply Theorem 2.4 of [7], thus |x 0 also is nonatomic) no analogous heredity results hold true in the case of charges concerning the predicates nonatomic and continuous as can be seen by direct arguments (for the nonatomic case, e.g., choose the field °U and the charge \x ' on °U as in the second part of the example in [16], p. 450. Let \x n be defined by \x n (B) = \(B 0 A") for any B E °U where X denotes the Lebesgue measure and A, = [0, 1/4 + 1 /(9 + n)] U (3/4 -1/(9 + AT), 1] for all nEX.…”

“…A. Sobczyk and P. C. Hammer have proved that any charge on a field °U can be decomposed uniquely into a continuous part and a part which can be written as a sum of at most two-valued charges on °IX ( [17], [16]). We…”

A New Proof of the Sobczyk-Hammer Decomposition Theorem

“…Proof. First we note that if 0 < µ ∈ ba(A) is not continuous, then there exists a two-valued 0 < λ ∈ ba(A) with λ ≤ µ (see Lemma in [3]). Moreover we can arrange λ so that it is 0 − m µ valued.…”