Baire Sets Research Articles

A Riesz representation theorem

Introduction. This paper deals with a Riesz representation theorem in the following setting: Let H be a compact Hausdorff space, let E and be locally convex Hausdorff topological vector spaces over the real or complex field. Let C(H, E) denote the continuous functions from H into E with the topology of uniform convergence. The purpose of this paper is to improve certain integral representation theorems for continuous linear transformations T from C(H, E) into The well-known Riesz representation theorem [10] gives a Stieltjes integral representation for T when H is a closed interval and when E and are the real numbers. There have been many generalizations of this theorem in the literature, and there have been two essentially different approaches giving rise to two different kinds of representation theorems. In one approach (see [1], [2], [5], [8], [9], [11]) y'T is written as an integral where y' is in the topological dual of The integral converges in the weak topology with a measure defined on the Borel sets with values in L(E, F), the space on continuous linear transformations of E into F. Under various additional assumptions on E, F, or T, T is written as an integral. (For example, this can be done if T is assumed to be compact or is assumed to be reflexive.) The most general of these theorems is due to Swong [11, Theorem 6.1, p. 283]. Another approach in the literature can be found in the papers [6], [12], [13], [14]. In these papers the method is of a constructive nature and T is written as an integral, and the convergence of the integral is in the e00 topology of F (norm topology in the case is a normed space), and T is thought of as a mapping of E into F by the canonical imbedding of into F. This is done by restricting the class of sets the measure is defined on. In [12] the class of sets is the intervals where H = [0, I], and E and are assumed to be normed spaces. In [6] and [13] the class of sets is the ring generated by compact Gj's. Recently in [14] Tucker and Wayment have been able to extend the measure to a larger proper subring of the Baire sets. In this paper it is shown that the restriction of the measure in [61, [12], [13], [14] was not necessary, and in fact T can be represented as an integral with convergence in the e topology with the measure defined on the Borel sets. This result thus implies and strengthens

On Product of Radón Measures

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.

Conditional Probability on $\Sigma$-Complete Boolean Algebras

Probability as measure on a Boolean algebra was presented by Kappos [5], but a treatment of conditional probability relative to a subalgebra is missing. The Stone space of a $\sigma$-complete Boolean algebra (see [10], p. 24) enables one to apply the concepts of conditional probability for a $\sigma$-algebra of subsets of some space (see [2], pp. 15-28), but the problem deserves closer attention. In this note we consider conditional probability with respect to a $\sigma$-subfield of the $\sigma$-field generated by the open-closed subsets of the Stone space of a Boolean $\sigma$-algebra. We show that there is always a regular conditional probability (see [4], p. 80) relative to a full $\sigma$-subalgebra of Baire sets. With a modified definition of probability on a Boolean algebra a treatment of conditional probability is possible without reference to the Stone space. For this a generalized integral is defined and the theory of integration is begun for it. A definition of conditional probability on a $\sigma$-complete Boolean algebra is given for which there is no regularity condition. We conclude the discussion with a study of the relationship of this theory with the conventional theory.

Baire sets in topological spaces

In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-Cech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrowka in [4]; second, the idea included in the proof of the theorem: “Every compact Baire set is aG δ ” as given inHalmos' text on measure theory [2; Section 51, theorem D].

Baire sets that are Borelian subspaces

The Borelian subspaces that are Baire sets were studied by Knowles & Rogers (1966). These sets are here called BB-sets and are characterized among all Borelian subspaces by the property of being separated from their complements under some continuous mapping into some separable metrizable space. The main properties of the BB-sets are developed by the methods of Frolík (1961). It is shown that every Borelian subspace of a space P is a Baire set in P if each open set of P is a Souslin set derived from the closed sets.

Descriptive sets

A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.

Baire sets and Baire measures

Baire sets and Baire measures

Von Neumann’s theorem on Abelian families of operators

The short proof below resembles but differs from von Neumann's original proof [2; 3]. Let W be the uniformly closed algebra generated by the set {I, An, A*, n, m= 1, 2, . }. Then, from the general theory of Banach algebras [I ] we see that S and C(9) are isometrically isomorphic and that the maximal ideal space 9Y of S is a compact metric space, since W is separable. Hence there is a mappingf: S-*9J of the Cantor set S onto W9. For t in [0, 1], let At(M) be the characteristic function of the set f([0, t)GS) C9M. Each of these sets, as the union of the compact sets f([o, t/n]CS) is a Borel, hence since 9I is metric, a Baire set. In accordance with the isometric isomorphism between the set of bounded Baire functions on SW and a super-ring of 2{ [i, 26 F, 26 G], Et(M) corresponds to a projection E(t), and clearly {E(t) I 0?t? 1 is a resolution of the identity. For B in 91, let b(t) be defined as follows:

Weak convergence of bounded sequences

PROOF. Assume that jjXn,jj ? AM, n = 1, 2, . It suffices to show that for each g in U*, lim g(x,,) =g(x). Now, U* is convex and compact (we only consider U* in its weak* topology) and the function Q defined by (Qx)(h)=h(x) for x in B, h in U*, is a linear isometry of B into C(U*), the space of continuous functions on U* with the supremum norm. Suppose that g is an element of U*. By the Bishopde Leeuw generalization [1, Theorem 5.6] of a well-known result of Choquet, there exists a a-ring of subsets of U* (generated by E and the Baire sets) and a non-negative finite measure , on this a-ring such that M(U*)=MA(E) and g(y)=f|uQyd,u for each y in B. Since IA(U*--E) =0, it follows that the functions Qxn converge to Qx almost everywhere with respect to ,u. Since I Qxnj | IIQxnII = IIXnI ? ,4!, we conclude from Lebesgue's bounded convergence theorem that g(x) = fQxd. = lim fQxndA = lim g(xn), which completes the proof.

Measures equivalent to the Haar measure

We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.

Jacobians for Measures in Coset Spaces

Let G be a locally compact topological group, let H be a closed subgroup and let G/H be the space of left cosets = xH with the natural topology. We denote by μ a non-negative measure in G/Hdefined on the ring of Baire sets. G acts by left multiplication as a transitive group of homeomorphisms on G/H: Every t ∈ G defines the homeomorphism We write, for E ⊂ G/H, tE = . The measure μ is called stable (cf. [3], [4]) if from t ∈ G, E ⊂ G/H and μ(E) = 0 follows μ(tE) = 0. We say that μ is locally finite [3], [5] if every set of positive measure contains a subset of positive finite measure.

A remark on a theorem of Beurling and Helson

The following result is due to Beurling and Helson [1, 2]: let , be a finite regular Borel measure on a locally compact abelian group G, and let u = Ean,uf7+v be its Lebesgue decomposition into discrete and continuous parts,' where /, is the mass 1 at g. Then if ,u has unimodular Fourier-Stieltjes transform (i.e., |,,=1), Z 2=1. The purpose of this note is to point out an extremely direct proof of the result, which also yields additional information. Let ,* represent, as usual, the measure defined by setting u*(f) =ff(-g)(dg), fGCo(G) or, alternatively, by setting 1u*(E)=Fp(E-') for all Baire sets E. Then ,u*= 2n w-,+V*, where v* is again continuous, and (u*) Thus | , =1implies (u*,*)^= | , 2-1= 'o and IA *,u* IAO. But

Baire Sets, Borel Sets and Some Typical Semi-Continuous Functions

Recently Hing Tong [3], M. Katētov [4] and C. H. Dowker [2] have established two sorts of insertion theorems for semi-continuous functions defined on normal and countably paracompact normal spaces. The purpose of this paper is to give the insertion theorem for some typical semi-continuous functions defined on a T-space. The relations between Baire sets and Borel sets in some topological spaces are also studied.

Remarks on a theorem of E. J. McShane

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if 7 is a family of open maps defined in one topological space X1 into another, X2, the total image 7(S) of a second category Baire set S in X1 has, under certain conditions on 7 and S, a nonvacuous interior. The point of these remarks is to show that his argument yields a theorem for a larger class than the second category Baire sets. From this there follow sligh4ly stronger and more specific versions of some of his results, including his principal theorem, as well as a proof that if S is a subset of a weak sort of topological group and S contains a second category Baire set, then the identity element lies in the interior of both S-1S and SS-1. There is also at the end an extension of Zorn's theorem on the structure of certain semigroups. In a topological space X let the closure and interior of a set E be denoted by E* and EO and the null set by A. For any set S let I(S) - U[GIG open, GnS is first category] and II(S)=X-I(S), and let III(S) be the open set II(S)0?I(X-S). By a fundamental theorem of Banach [2], SnI(S)* is first category and hence S is second category if and only if II(S)O7A. From these we note that if N is a non-null open subset of III(S), then N-S is in the first category set (X - S)G'iI(X - S), and NnQS cannot be first category since N is non-null open and disjoint with I(S). This gives us the following lemma. LEMMA 1. For any non-null open subset N of III(S), the sets N- S