Weak Limits of Measures and the Standard Part Map

Abstract

A construction is given, using the standard part map st, of the weak* standard part of an internal Baire measure in the nonstandard extension of a compact Hausdorff space.It is shown that the inverse image with respect to st of a Borel set is universally measurable with respect to completions of the o-algebra generated by internal Baire sets.Applications and extensions of these results to noncompact spaces are given.In this note, we give a construction of the weak* standard part for a finite-valued, internal Baire measure in the nonstandard extension *X (see [15] or [17]) of a compact Hausdorff space X.The construction uses the measurability of the standard part map st: *X ^ X; i.e., for each Baire set B c X, st~x(B) is in the external a-algebra o(*9>(X)) generated by the internal Baire sets in *X.We shall also show that for each Borel set A c X, st~x(A) is "universally" measurable, i.e., st~x(A) is in the completion of o(*9>(X)) with respect to any internal, finite-valued Baire measure on *X.These results have extensions to noncompact spaces and are related, as we shall show, to results of Anderson and Rashid in [3].Let (X, ') be a compact Hausdorff space, C(X) the continuous real-valued functions on X, <$ (X) the class of Baire subsets of X, and M(X) the space of signed Baire measures nonl with finite norm \\\\=\p\(X) = p + (X) + ^(X).Fix a -saturated enlargement [11] of a structure containing X and the real numbers R, with k > max(N0, card T).Let o(*%(X)) be the smallest ordinary a-algebra of subsets of *X containing the class *%(X) of internal Baire sets in *X.We let *M[X) denote the set of v E *M(X) with (|M|) < + oo.Given v E *MA[X), let p0(A) = (v(A)) for each A E *<$>(X).By Theorem 1 of [8], v0 has a unique, countably additive, real-valued extension i>0 defined on o(*% (X)).We shall use v0 to construct the weak* standard part of v, i.e., the unique /j, M(X) such that for each/ E C(X),