Continuous Selection of Representing Measures by H. S. Bear

where Q is a continuous function of x and x—>Q(x, •) is continuous with respect to the £»(/*) norm. If it is also the case that ^ | T is dense in Li(ju)> then ()(•, 0) is a limit (uniform convergence on compact subsets of A) of functions in B. These results also give integral representations like (1) and (2) for a complex function algebra, simply by considering the space B of real parts of the algebra. The details of this work will appear in [3]. We use the following notation throughout this paper: X is a compact Hausdorff space, with topology 3. B is a linear subspace of CR(X), containing the constant functions, and separating the points of X. T is the Silov boundary of B in X. BA, for any set AQX, is the closure of B\ A in the topology of uniform convergence on compact subsets of A. B+(A, z) = {u\A:uEB, U>0, u(z) = l}. (JBA is an abstract version of "all harmonic functions on A," and J3(A, z) is the set of normalized-at-z positive i3-functions.)