Solovay Model and Duality Principle Between the Measure and the Baire

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P if the sentence P is equivalent to the sentence P ∗ obtained from the sentence P by interchanging in it the notions of the above small sets. The present paper is devoted to study of the question asking whether the duality principle holds true between the measure and the Baire category with respect to the sentence - ” The domain of an arbitrary generalized integral for a vector-function is of ﬁrst category .” The paper is organized as follows. In Section 2 we consider some auxiliary notions and facts from the measure theory and topology. In Section 3 we give the proof of the main result.

2. Some auxiliary notions and facts from measure theory and topology Definition 2.1. Let (E, τ) be a topological space. We say that a subset Y ⊆ E has Baire property if Y admits the following representation ′ ′′ Y = (Z ∪ Z ) \ Z , ′ ′′ where Z is an open set in E, Z and Z are sets of the first category. Definition 2.2. We say that a topological space (E, τ) is Baire space if an arbitrary non-empty open subset of E is the set of second category. Remark 2.3. Following Solovay [1], let ’ZF ’ denote the axiomatic set theory of Zermelo-Fraenkel and let ’ZF + DC’ denote the system obtained by adjoining a weakened form of the axiom of choice, DC, (see p. 52 of [1] for a formal statement of DC). The system ZF + DC is important because all the positive results of elementary measure theory and most of the basic results of elementary functional analysis, except for the Hahn-Banach theorem and other such consequences of the axiom of choice, are provable in ZF + DC. In particular, the Baire category theorem for complete metric spaces and the closed graph theorem for operators between Fréchet spaces are provable in ZF + DC. Let ZF C be Zermelo-Frankel set theory together with the axiom of choice. In the presence of the axiom of choice, we identify cardinals with initial ordinals. A cardinal ℵ is regular, if each order unbounded subset of ℵ has power ℵ. A cardinal ′ ′ ℵ is inaccessible, if it is regular, uncountable, and for ℵ < ℵ, 2ℵ < ℵ. Let I be the statement: ”There is an inaccessible cardinal”. Lemma 2.4. ([1], THEOREM 1, p.1) Suppose that there is a transitive ǫ-model of ZF C + I. Then there is a transitive ǫ-model of ZF in which the following pro positions are valid . (1) The principle of dependent choice (DC). (2) Every set of reals is Lebesgue measurable (LM). (3) Every set of reals has the property of Baire. (4) Every uncountable set of reals contains a perfect subset. (5) Let {Ax : x ∈ R} be an indexed family of non-empty set of reals with index set the reals. Then there are Borel functions h1,h2 mapping R into R such that (a) {x : h1(x) ∈/ Ax} has Lebesgue measure zero. (b) {x : h2(x) ∈/ Ax} is of first category. Let (X,S,µ) be a probability space and H a separable Banach space over the vector field R. Let denote by H(X,S,µ) a class of all µ-measurable mappings of the SOLOVAYMODELANDDUALITYPRINCIPLE 3

X into H. As usual, we will identify equivalent mappings, i.e., we will consider the factor-space with respect to an equivalent relation on H(X,S,µ). If µ is separable and the metric ρ in H(X,S,µ) is deﬁned by

ρ(f,g)= ||f − g||/(1 + ||f − g||)dµ, ZX then H(X,S,µ) stands a Polish topological vector space without isolated points. Suppose that µ is separable and diffused. Let M be a vector subspace of H(X,S,µ) which contains the class of all measurable step functions. A linear operator I : M → H is called a generalized integral for a vector-function if for each step function g : X → H the equality n I(g)= aiµ(Xi), Xi=1 holds true, where (Xi)1≤i≤n is a measurable partition of X and g takes the value ai at points of Xi for each i(1 ≤ i ≤ n). It is obvious that Reiman, Lebesgue and Danjua integrals are partial cases of the generalized integral for vector-function and they are realized when under H is taken the real axis R considered as a Banach space over vector field R. Let denote by (SM) the transitive ǫ-model of ZF which comes from Lemma 2.4. Lemma 2.5. ([5], Theorem, p. 34 )(SM) The domain of an arbitrary generalized integral for a vector-function is of first category in H(X,S,µ). Definition 2.6. Let V be a Polish topological vector space. Let K be the class of all probability diffused Borel measures defined on the Borel σ-algebra B(V). We denote by B(V)µ the completion of B(V) with respect to the measure µ for µ ∈ K. µ A set E ⊂ V is called universally measurable if E ∈ ∩µ∈K B(V) . Lemma 2.7. ( [9], Remark 3.4, p. 40 ) (SM) Let V be a complete separable metric space. Then every subset of V is universally measurable. Definition 2.8 ( [7], Definition 1, p. 221). A measure µ is said to be transverse to a universally measurable set S ⊂ V if the following two conditions hold: (i) There exists a compact set U ⊂ V for which 0 <µ(U) < 1. (ii) µ(S + v) = 0 for every v ∈ V, where µ denotes a usual completion of the measure µ. Definition 2.9 ( [7], Definition 2, p. 222). A universally measurable set S ⊂ V is called shy if there exists a measure transverse to S. More generally, a subset of V is called shy if it is contained in a universally measurable shy set. The complement of a shy set is called a prevalent set. Remark 2.10. Note that the class of all shy sets in a olish topological vector space ′′ V constitutes a σ-ideal(see, [7], Fact 3 , p. 223).

3. Main result Theorem 3.1. In the model (SM) the domain of an arbitrary generalized integral for a vector-function in H(X,S,µ) is shy. Proof. Let I be a generalized integral for a vector-function in H(X,S,µ) and M be it’s domain. Since H(X,S,µ) is a Polish topological vector space, by Lemma 2.7 we know that M is universally measurable. By Lemma 2.5 we know that M is the set 4 G. PANTSULAIA of the ﬁrst category in H(X,S,µ). By Remark 2.3 we know that H(X,S,µ) is the Baire space which implies that H(X,S,µ) \ M 6= ∅. Let v ∈ H(X,S,µ) \ M. Let us show that v spans a line L such that every translate of L meets M in at most one point. Indeed, assume the contrary. Then there will be an element z ∈ H(X,S,µ) and two diﬀerent parameters t1,t2 ∈ R such that z + t1v ∈ M and z + t2v ∈ M. Since M is a vector subspace of H(X,S,µ) we deduce that (t2 − t1)v ∈ M. Using the same argument we claim that v ∈ M because t2 −t1 6= 0, but the latter relation is the contradiction. Hence the Lebesgue measure supported on L is transverse to a universally measurable set M which means that M is shy. This ends the proof of the theorem.

By using Lemma 2.5 and Theorem 3.1 we get Theorem 3.2. In the model (SM) the duality principle between the measure and the Baire category is valid with respect to the sentence P defined by ” The domain of an arbitrary generalized integral for a vector-function is of first category in H(X,S,µ).” Acknowledgement. The main result of the present paper was reported on the Section of Foundations of Mathematics and Mathematical Logic of the XXIX Enlarged Sessions of the Seminar of Ilia Vekua Institute of Applied Mathematics of Ivane Javakhisvili Tbilisi State University. References [1] Solovay R.M., A model of set theory in which every set of reals is Lebesgue measurable,Ann.Math.,92, 1970, 1–56. [2] Shelah, S. Can you take Solovay’s inaccessible away ?. Isr. J. Math. 48 1984), 1–47 [3] Raisonnier, Jean. A mathematical proof of S. Shelah’s theorem on the measure problem and related results. Israel J. Math. 48(1) (1984), 48–56. [4] Wright, J. D. Maitland. All operators on a Hilbert space are bounded. Bull. Amer. Math. Soc. 79 (1973), 1247–1250. [5] Pantsulaia G.R., Generalized integrals, Soobshch. Akad. Nauk Gruzin. SSR., 117(1) (1985), 33–36 (in Russian). [6] Pantsulaia G.R., Relations between shy sets and sets of νp-measure zero in Solovay’s Model, Bull. Polish Acad. Sci., 52(1) (2004). [7] Hunt B.R., Sauer T., Yorke J.A., Prevalence:A Translation-Invariant “Almost Every” On Infinite-Dimensional Spaces, Bulletin (New Series) of the American Math- ematical Society, 27(2) (1992), no. 10, 217–238. [8] Oxtoby J.C., Measure and Category, Berlin, Springer (1970). [9] Pantsulaia G., Generators of Shy Sets in Polish Groups. Nova Science Publishers, Inc., New York, 2011

I. Vekua Institute of Applied Mathematics, Tbilisi State University, Tbilisi DC 0143, Georgia E-mail address: [email protected]