Countably determined sets and a conjecture of C.W. Henson

Introduction

“…Since ś U XpM i q is closed in N, there is ǫ ą 0 such that dprpa i qs U , ś U XpM i qq ą ǫ. Let ϕ and δ ą 0 be as in (4). Since XpNq Ď Zpϕ N q, we have that ϕ M i pa i q ă δ for U-almost all i P I, hence dpa i , XpM i qq ď ǫ for U-almost all i P I, which is a contradiction.…”

“…To see that (4) implies ( 5), we first show that ś U XpM i q Ď XpNq. To this end, fix ǫ ą 0 and let ϕ and δ be as in (4). If rpa i qs U P ś U XpM i q, then ϕ M i pa i q " 0 for all i P I, hence ϕ N pra i s U q " 0 and thus dprpa i qs U , XpNqq ď ǫ.…”

Borel Measure Extensions of Measures Defined on Sub-σ-Algebras

“…Let X be a C 8 ech-complete space, let A/Bo(X) be a _-algebra, and let & be a measure on A such that & A is {-smooth. Since X is C 8 echcomplete, there exists a sequence V n of open coverings of X with ns( C X)= n V # V n C V (Theorem 5.1 of [Re2]). Fix a standard =>0.…”

“…But we have not been able to determine whether arbitrary C 8 ech-complete spaces are thick. By Theorem 5.1 of [Re2], given a C 8 ech-complete space X, there exists a sequence of open coverings V n of X with…”

Pushing down infinite Loeb measures

“…The question of when st is measurable has been extensively studied; see [4] and [9] for early papers on the question, and the discussion following Theorem 3.2 of [16] for more recent results. The reader is also referred to [8], [2], [11], [12], as well as to [5] and [6] for a functional approach.…”