Algebra Trigonometry

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GN 0 3 is seen by applying quotient spaces as in the previous theorem. One has to consider spaces 6 (Xi)= C (Xi) /JR. 1. All the notions and results in this section are taken from Y. Sternfeld's papers (133] and (134]. §4. CONSTRUCTING FUNCTION FAMILIES SEPARATING BOREL MEASURES 29 §4. Constructing function families separating Borel measures Let us first agree on some convenient terminology and notation.

22). Consider one more space. Let c0 (T) denote the class of bounded real-valued functions x(t) defined on an {arbitrary) set T such that for any c > 0 the set {t ET: lx(t)I > c} is finite. c0 (T) is a closed subspace in B(T), and c0 (T)* = £1 {T) {[35, Chapter II, Section 2]). We say that a map cp : X ---+ Y has finite rank if the full preimage of any point y E Y is a finite set. The system of mappings cpi : x---+ xi. i = 1, ... , has finite rank if each mapping cpi has finite rank. 2 hold in relation to X, {Xi}, {cpi}, {hi}, and let the system of mappings {cpi} have finite rank.

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