Weak limits of measures and the standard part map

Abstract

A construction is given, using the standard part map st, of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="weak Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mtext>weak</mml:mtext> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {weak}^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra generated by internal Baire sets. Applications and extensions of these results to noncompact spaces are given.