Note on the spacesP(S) of regular probability measures whose topology is determined by countable subsets

Abstract

Two closely connected topics are discussed: countable tightness in the spaces P(S) of regular probability measures with the weak topology and a convex analogue to Lindelof property of the weak topology of the function spaces C(S) defined by H. H. Corson. The main result of this note exhibits a rather wide class of compact spaces stable under standard operations including the operation P(S) 9 such that within this class both of the properties we deal with are dual each other and they behave in a regular way. Some related open problems are stated.1* Introduction* In this note we consider two closely connected topics: countable tightness in the spaces P(S) of regular probability measures on compact spaces endowed with the weak* topology and property (C)-a convex analogue to Lindelof property of the weak topology of function spaces C(S) defined by H. H. Corson [6] (for the terminology and definitions see § §2 and 3).Our results are related to the following two problems: C(S) is equivalent to a property of P(S) which is a convex analogue to countable tightness (see Lemma 3.2). This property is a priori weaker than countable tightness but no example known to us shows that this is realy the case. So, for what compact spaces S countable tightness of P(S) is equivalent to property (C) of C(S), or putting this another way, when property (C) and countable tightness are dual each other'! (B) Does the function space C(S x S) or C(P(S)) have property (C) provided that the space C(S) has this property! Does countable tightness of the space P(S x S) or P(P(S)) follow from countable tightness of the space P(S)?It should be mentioned here that the only examples we know of compact spaces S with countable tightness for which C(S) fails to have property (C) or P(S) fails to have countable tightness, due to Hay don [14] and to van Douwen and Fleissner [7], are constructed under additional set theoretic hypotheses. This yields yet another problem, whether in such examples some extra axioms for set theory are necessary (the results of this note, however, have no connection to this question). 185 ROMAN POLWe shall consider the class of compact spaces S such that each regular measure on S is determined by its values on a countable collection of compact sets (Definition 3.3). Our main result (see Theorem 4.1 and Corollary 4.2) is that this class of spaces is closed under the operation P(S) and some other standard topological operations, and that in the realm of this class the questions stated in (A) and (B) have always a positive answer.In this context the question arises how wide is the class of spaces we deal with and to what extent the countable determinantness of regular measures on S is connected to property (C) of C(S) or to countable tightness of P(S)? The class we consider includes many "classical" compact spaces (cf. Example 3.7). In fact, no example is known to us of a compact space S outside of this class for which C(S) has property (C) or P(S) has countable tightness (but we see also no reason why such spaces should not exis...

On Corson compacta and embeddings of 𝐶(𝐾) spaces

“…The theorem follows directly from Lemma 2.1(c) and the following lemma, which is due to Pol [24], Lemma 4.4; we enclose a (slightly different) proof of the latter.…”

“…for every open U ⊆ K; see Pol [24] and Mercourakis [18]. It will be convenient to say, whenever ( †) holds, that F approximates U from below (with respect to some μ ∈ P (K), which is clear from the context).…”

“…The second part can be checked in a standard way; see Pol [24], Lemma 3.6. Now we have come to the main point of the section.…”

Compact subsets of the first Baire class

“…It should be noted that Lemma 8 can also be deduced from the main result of Krawczyk [Kr], the paper that contains the first substantial progress towards the problem of Pol [Po1].…”

Pettis integration via the Stonian transform

“…Thus neighborhoods of points (7,0) (resp., (t, 1)) look to the left (resp., right). K is compact Γ 2 , first countable, separable, and perfectly normal, but not metrizable (see [20] for additional information). If 0 < t < 1, letΛ, = {(/>, j): p < /}, B t = A t U {(/, 0)}, Q = B t U {(/, 1)}.…”