GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 3, 1999, 201-212

ON UNCOUNTABLE UNIONS AND INTERSECTIONS OF MEASURABLE SETS

M. BALCERZAK AND A. KHARAZISHVILI

Abstract. We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are considered in detail, and it is shown that the Lebesgue measure and the Baire property differ essentially in this aspect. Another difference between measure and category is illustrated in the case of some uncountable intersections of sets of full measure (comeager sets, respectively). We also discuss a topological form of the Vitali covering theorem, in connection with the Baire property of uncountable unions of certain sets.

0. Introduction

A mathematician working in probability theory, the theory of random processes or in various fields of modern analysis is quite frequently obliged to show that the union ⋃{X_{a} : a ∈ A} or the intersection ⋂{X_{a} : a ∈ A} of a given uncountable family {X_{a} : a ∈ A} of measurable sets is measurable provided that the sets X_{a} have some additional “good” structure. Here the measurability is meant in the sense of the respective σ-algebra S of subsets of a given nonempty basic set E containing each set X_{a} (a ∈ A). For instance, let E coincide with the real line R and let S be the σ-algebra of all Lebesgue measurable subsets of R. Let us consider the family T of all S-measurable sets X ⊆ R such that each point x from X is a density point of X. It is well known (see, e.g., [1], Chapter 22) that T forms a topology which is usually called the density topology on the real line. Thus the union of each subfamily of T belongs to T , and hence is S-measurable. In particular, we see that uncountable unions of S-measurable sets belonging

1991 Mathematics Subject Classification. 28A05, 28A12, 54E52. Key words and phrases. Lebesgue measure, Jordan measure, Baire property, Vitali

covering, Jordan curve, Suslin condition, simplex, meager set.

201

1072-947X/99/0500-0201$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

to T are S-measurable, too. The same situation holds for the von Neumann-Maharam topology which is a generalization of the density topology and can be introduced for an arbitrary measure space (E, S, μ) where E is a nonempty basic set, S is a σ-algebra of subsets of E and μ is a nonzero σ-finite complete measure defined on S (see [1], Chapter 22). Note that a number of interesting examples and questions concerning the measurability of uncountable unions of measurable sets are discussed in [2].

The main aim of this paper is to illustrate that the phenomenon producing measurable unions (or intersections) for uncountable families of “good” measurable sets depends on very delicate conditions. Another goal is to express some similarities and differences between measure and category in that aspect. In Section 1, we briefly consider the most classical situation dealing with an arbitrary family {S_{t} : t ∈ T } of closed n-simplexes in the n-dimensional Euclidean space R^{n}. Applying some well-known properties of the standard Lebesgue and Jordan measures in R^{n} (see, e.g., [3]), we show that the set ⋃{S_{t} : t ∈ T } is Lebesgue measurable and has the Baire property. In Section 2, we are concerned with arbitrary families of sets that are homeomorphic to a closed n-simplex in R^{n} (such sets are called topological n-simplexes). It turns out that, in this situation, the measure and the category cases differ essentially. Namely, an arbitrary family of topological n-simplexes has the union with the Baire property. On the other hand, there exists a family of topological n-simplexes in R^{n} (n ≥ 2) with the union nonmeasurable in the Lebesgue sense. Therefore, from this point of view, the Baire property behaves better than Lebesgue measure. Section 3 witnesses that in some situations it can happen conversely, that is, measure behaves better than the Baire property. Namely, in connection with a theorem of Goldstern [4] concerning certain uncountable intersections of coanalytic sets with the full probability measure, we show that an analogous result fails to be true in the category case. In Section 4, we introduce the notion of a topological Vitali space. It turns out that these spaces characterize the situation where, for a wide class of sets (called admissible), the union of an arbitrary subfamily is countably approximable and consequently has the Baire property.

Our notation and terminology concerning set theory, general topology, measure and category are fairly standard. In particular, we denote:

ω – the set of all natural numbers, i.e., ω = {0, 1, 2, …, n, … }; μ – a measure on a given basic space (as a rule, we assume that μ is

nonzero, σ-finite and complete); dom(μ) – the family of all μ-measurable sets; ν_{n} – the classical Jordan measure in R^{n}; λ_{n} – the classical Lebesgue measure in R^{n}. Let E be an arbitrary topological space. If X ⊆ E, then cl(X), int(X)

and bd(X) stand for the closure, the interior and the boundary of X, respectively. We denote:

NW D(E) – the ideal of all nowhere dense subsets of E;

K(E) – the σ-ideal of all meager (i.e., first category) subsets of E; Br(E) – the σ-algebra of all subsets of E with the Baire property. Recall that a set X ⊆ E has the Baire property iff X can be represented

as the symmetric difference X = U△P where U is open and P is meager in E. Moreover, if X ⊆ E can be expressed in the form X = U△P where U is open and P is nowhere dense in E, then X is called an open set modulo a nowhere dense set (cf. [5], §8, V) or, briefly, an open set modulo NW D(E).

We shall say that a subset X of a topological space E is admissible if

X ⊆ cl(int(X)).

Clearly, every open set is admissible. If X is a regular closed set (i.e., X = cl(int(X))), then X is admissible, as well. In addition, if X is admissible, then from the relations

int(X) ⊆ X ⊆ cl(int(X)), cl(int(X)) \ int(X) ∈ NW D(E)

it follows that X is open modulo a nowhere dense set (in general, the converse is false).

In our further considerations, the notion of a Vitali covering plays an essential role. So, we introduce this notion for a general topological space and for an arbitrary subset of that space. We say that a family V of subsets of a topological space E forms a Vitali covering of a given set X ⊆ E if, for each point x ∈ X and for each neighbourhood U of x, there exists a set V ∈ V such that x ∈ V ⊆ U. Consequently, if V is a Vitali covering of X, then the family of sets {V ∩ X : V ∈ V} forms a net for the subspace X of E. As a rule, the Vitali coverings considered below are assumed to consist of admissible sets. We also recall that a topological space E satisfies the Suslin condition (the countable chain condition) if each disjoint family of nonempty open sets in E is countable.

1. Unions of n-Simplexes

By an n-simplex we mean a closed nondegenerate simplex in the Euclidean space R^{n}. If a real α > 0 is fixed, we say that a bounded set X ⊆ R^{n} is α–regular if λ_{n}(X) ≥ αλ_{n}(V (X)) where V (X) denotes a closed ball with the minimal diameter, for which we have the inclusion X ⊆ V (X) (clearly, in this definition a ball can be replaced by a cube). Note that every n-simplex is an α–regular set for some α > 0.

Theorem 1.1. Let {S_{t} : t ∈ T } be an arbitrary family of n-simplexes in R^{n}. Then the set X = ⋃{S_{t} : t ∈ T } is Lebesgue measurable and possesses the Baire property.

Proof. Obviously, the set X can be expressed in the form

X = ⋃_{m∈ω, m>0} ⋃_{t∈T_{m}} S_{t}

where

T_{m} = {t ∈ T : S_{t} is 1/m−regular}.

It suffices to show that, for a fixed integer m > 0, the set

X_{m} = ⋃{S_{t} : t ∈ T_{m}}

is Lebesgue measurable. For any t ∈ T_{m}, x ∈ S_{t} and c ∈ ]0, 1[, let S_{t}^{c}(x) denote the image of S_{t} under the homothetic transformation

y → x + c(y − x) (y ∈ R^{n}).

Observe that the family of n-simplexes

F_{m} = {S_{t}^{c}(x) : t ∈ T_{m} & x ∈ S_{t} & c ∈ ]0, 1[}

forms a Vitali covering of X_{m}. Additionally, F_{m} consists of simplexes which are 1/m–regular sets. Thus, by the generalized Vitali theorem (see [6], Chapter 4, Theorem 3.1), there exists a countable disjoint family F_{m}^{∗} ⊆ F_{m} such that λ_{n}(X_{m} \ ⋃ F_{m}^{∗}) = 0. Evidently, ⋃ F_{m}^{∗} is Lebesgue measurable. Also, we have ⋃ F_{m}^{∗} ⊆ X_{m} by the definition of F_{m}. Hence X_{m} is Lebesgue measurable.

Our argument in the category case also uses measure-theoretical tools. Namely, we apply the Jordan measure ν_{n} in R^{n} and the respective inner measure (ν_{n})_{∗}. By the classical criterion, a bounded set Y ⊆ R^{n} is Jordan measurable iff ν_{n}(bd(Y)) = 0 where bd(Y) stands for the boundary of Y. Since bd(Y) is compact, the equality ν_{n}(bd(Y)) = 0 implies that bd(Y) is nowhere dense. Consequently, every Jordan measurable set Y can be expressed as the union of the open set int(Y) and the nowhere dense set Y ∩ bd(Y). In particular, we see that Y possesses the Baire property.

In view of the above remark, it suffices to express our set X as a countable union of Jordan measurable sets. For an arbitrary integer m > 0, we denote

B_{m} = {x ∈ R^{n} : ||x|| ≤ m}, G_{m} = {S_{t} : t ∈ T & S_{t} ⊆ B_{m}},

where ||x|| stands for the usual Euclidean norm of x ∈ R^{n}. We obviously have

X = ⋃_{m∈ω, m>0} ⋃ G_{m}.

Now, let us consider the inner Jordan density of an arbitrary set Z ⊆ R^{n} at a point x ∈ R^{n}, given by the formula

d_{∗}(Z, x) = inf{((ν_{n})_{∗}(Z ∩ Q))/ν_{n}(Q) : Q ∈ Q(x)},

where Q(x) stands for the family of all closed cubes with centre x and with diameters ≤ 1. According to the well-known result (see [3], Chapter 3), if there exists ε > 0 such that, for each point z of a bounded set Z ⊆ R^{n}, we have d_{∗}(Z, z) ≥ ε, then Z is Jordan measurable. Observe that

G_{m} = ⋃_{k∈ω, k>0} G_{m}^{k}

where

G_{m}^{k} = {S_{t} ∈ G_{m} : (∀x ∈ S_{t})(d_{∗}(S_{t}, x) ≥ 1/k)}.

Then the set ⋃ G_{m}^{k} is bounded and d_{∗}(⋃ G_{m}^{k}, x) ≥ 1/k for each point x ∈ ⋃ G_{m}^{k}. Consequently, ⋃ G_{m}^{k} is Jordan measurable. We thus see that the set X is expressible as a countable union of Jordan measurable sets. Hence it has the Baire property.

We want to finish this section with several simple remarks concerning the theorem just proved. First of all let us note that the union of a family of n-simplexes may have a rather bad descriptive structure. Indeed, suppose that n ≥ 2 and consider some hyperplane Γ in the space R^{n}. Let Y be any subset of Γ. It is easy to construct a family of n-simplexes in R^{n}, such that the intersection of its union with the hyperplane Γ coincides with the set Y. Consequently, if Y is not λ_{n−1}-measurable or does not possess the Baire property in Γ, then the above-mentioned union is not an analytic (coanalytic) subset of R^{n}. In a similar way one can construct a family of n-simplexes whose union is not a projective subset of R^{n}.

It immediately follows from Theorem 1.1 that the union of an arbitrary family of convex bodies in R^{n} is λ_{n}-measurable (because any convex body in R^{n} can be represented as the union of a family of n-simplexes). Analogously, the union of an arbitrary family of convex bodies in R^{n} has the Baire property. The latter fact remains true for any family of convex bodies in a topological vector space (see, e.g., Theorem 2.1 below containing a more general result).

2. Unions of Topological n-Simplexes

By a topological n-simplex we mean a topological space homeomorphic to an n-simplex. Our purpose in this section is twofold. First, we are going to show that the union of an arbitrary family of topological n-simplexes in R^{n} has the Baire property. Next, we shall prove that, for n ≥ 2, an analogous statement fails to be true if the possession of the Baire property is replaced by Lebesgue measurability. The first of these two results will be derived from the following general theorem:

Theorem 2.1. Let E be an arbitrary topological space and let F be a family consisting of admissible subsets of E. Then ⋃ F is open modulo NW D(E). In particular, ⋃ F has the Baire property.

Proof. We put V = ⋃{int(F) : F ∈ F}. Since F consists of admissible sets, we have, for each F ∈ F,

F ⊆ cl(int(F)) ⊆ cl(V).

Thus, we obtain ⋃ F ⊆ cl(V) and, therefore,

V = ⋃{int(F) : F ∈ F} ⊆ ⋃ F ⊆ cl(V).

Hence ⋃ F = V ∪Z for some set Z ⊆ cl(V)\V, which gives the assertion.

Corollary 2.1. For each integer n ≥ 1 and for each family of topological n-simplexes {S_{t} : t ∈ T } in R^{n}, the set ⋃{S_{t} : t ∈ T } has the Baire property.

Obviously, Corollary 2.1 yields another (purely topological) proof of the category part of Theorem 1.1.

The next statement shows that the measure case for unions of topological simplexes is completely different.

Theorem 2.2. For each integer n ≥ 2, there exists a family {Z_{t} : t ∈ T } of topological n-simplexes in R^{n} such that the set ⋃{Z_{t} : t ∈ T } is not measurable in the Lebesgue sense.

Proof. It is enough to consider the case n = 2 since, if a family {Z_{t} : t ∈ T } satisfies the assertion of our theorem for n = 2, then the family {Z_{t} ×[0, 1]^{n−2} : t ∈ T } satisfies the assertion of the theorem for an arbitrary integer n > 2. So, we restrict our further consideration to the case n = 2. In the sequel, by a Jordan curve we mean a homeomorphic image of the unit circle. It is well known that there exists a Jordan curve L in R^{2} possessing a positive two-dimensional Lebesgue measure, i.e., λ_{2}(L) > 0. The construction of L can be done directly. Another idea is to derive from the Denjoy-Riesz theorem (see, e.g., [5], §61, V, Theorem 5) that each compact zero-dimensional set C in R^{2} is contained in a Jordan curve. Taking as C a Cantor-type set in R^{2} with λ_{2}(C) > 0, we get the desired curve L. Now, by the Jordan Curve Theorem, R^{2} \ L has exactly two components: one bounded and one unbounded. The bounded component will be denoted by U. We shall use the Schönflies theorem (see [5], §61, II, Theorem 11) according to which, for any points x ∈ L and y ∈ U, there is a simple arc l, with end-points x and y, such that l \ {x} ⊆ U. In fact, a bit sharper version is needed where the simple arc l is a quasi-polygonal curve, i.e., the set l \ {x} consists of countably many linear segments which converge (in the Hausdorff metric) to {x}. (Cf., e.g., [7], Appendix to Chapter IX.)

Now, let us pick a Lebesgue nonmeasurable set L^{∗} ⊆ L. (The existence of L^{∗} is well known - see, e.g., [1], Chapter 5.) We shall define a family {Z_{x,y} : x ∈ L^{∗}, y ∈ U} of sets lying in the plane and satisfying the following conditions:

(1) Z_{x,y} is homeomorphic to a closed nondegenerate triangle;

(2) Z_{x,y} \ {x} ⊆ U;

(3) x and y are boundary points of Z_{x,y}.

So, fix x ∈ L^{∗} and y ∈ U and pick a point y′ such that the segment [y, y′] is contained in U. By the above-mentioned sharp version of the Schönflies theorem, we choose quasi-polygonal curves P_{x,y} (joining x and y) and P_{x,y′} (joining x and y′) such that P_{x,y} \ {x} ⊆ U, P_{x,y′} \ {x} ⊆ U. First, by a simple modification, we can ensure that P_{x,y} ∩ [y, y′] = {y}, P_{x,y′} ∩ [y, y′] = {y′}. Next, using the fact that the segments of P_{x,y} and of P_{x,y′} converge to the point {x}, we modify P_{x,y} and P_{x,y′} (step by step) to ensure that P_{x,y} ∩ P_{x,y′} = {x}. If it is done, the set L_{x,y} = P_{x,y} ∪ [y, y′] ∪ P_{x,y′} forms a Jordan curve. We define Z_{x,y} as the closure of the bounded component of R^{2} \ L_{x,y}. Finally, we put t = (x, y), Z_{t} = Z_{x,y}, T = L^{∗} × U. Then the family {Z_{t} : t ∈ T } satisfies the assertion of the theorem. Indeed, each set Z_{t} (t ∈ T) is homeomorphic to a closed nondegenerate triangle (this is a strong version of the Jordan theorem for the plane, which does not hold for an Euclidean space of higher dimension). Since the set ⋃{Z_{t} : t ∈ T } \ U = L^{∗} is Lebesgue nonmeasurable, the set ⋃{Z_{t} : t ∈ T } is Lebesgue nonmeasurable, too.

3. Uncountable Intersections of Thick Sets

Here we consider the following problem concerning uncountable intersections of measurable sets. Let S be a σ-algebra of subsets of a given nonempty set E and let I be a σ-ideal of subsets of E such that I ⊆ S. In such a case, the triple (E, S, I) is called a measurable space with a σideal. Additionally, let the pair (S, I) fulfil the countable chain condition (in short, ccc), which means that each disjoint subfamily of S \ I is countable. One can ask the question what should be assumed, for an uncountable family {X_{t} : t ∈ T } of subsets of E, such that (∀t ∈ T)(E \ X_{t} ∈ I), to get the relation E \ ⋂{X_{t} : t ∈ T } ∈ I. The main difficulty is to have ⋂{X_{t} : t ∈ T } ∈ S. It seems natural to introduce the set

W = {(t, x) ∈ T × E : x ∈ X_{t}},

and thus the sets X_{t} are equal to the vertical sections W_{t} = {x ∈ E : (t, x) ∈ W } of W. So, our problem can now be formulated as follows. Let W be a set of some “good” structure in the product set T × E, with thick sections W_{t}, t ∈ T, (i.e., E \ W_{t} ∈ I). How is it possible to obtain the relation E \ ⋂{W_{t} : t ∈ T } ∈ I?

In connection with the general question posed above, let us mention especially the result of Goldstern [4] which deals with a particular (but important for various applications) case. Namely, let E be a perfect Polish space with a probability Borel measure μ. The completion of μ is denoted by μ̄. Then S and I stand for the σ-algebra of μ̄-measurable sets and the σ-ideal of μ̄-null sets (respectively). It is well known that (S, I) fulfils the ccc. The role of T is played by the zero-dimensional perfect Polish product space ω^{ω} where ω is equipped with the discrete topology. A natural order in this product space is defined by t ≤ t′ ⇔ (∀n ∈ ω)(t(n) ≤ t′(n)) for any t and t′ from ω^{ω}.

M. Goldstern proved the following result (see [4], Lemma 6).

Theorem 3.1. Assume that W ⊆ ω^{ω} × E is a coanalytic set satisfying the conditions:

(a) (∀t ∈ ω^{ω})(μ̄(W_{t}) = 1);

(b) (∀t, t′ ∈ ω^{ω})(t ≤ t′ ⇒ W_{t} ⊇ W_{t′}). Then μ̄(⋂{W_{t} : t ∈ ω^{ω}}) = 1.

We are going to show that the category analogue of Goldstern’s theorem is false. Now, we have that E = [0, 1], the σ-algebra S consists of subsets of E with the Baire property and the σ-ideal I consists of all meager subsets of E. It is well known that the pair (S, I) fulfils the ccc.

Theorem 3.2. There is an open set W ⊆ ω^{ω} × [0, 1] such that

(a) (∀t ∈ ω^{ω})(W_{t} is comeager in [0, 1]),

(b) (∀t, t′ ∈ ω^{ω})(t ≤ t′ ⇒ W_{t} ⊇ W_{t′}),

(c) the set ⋂{W_{t} : t ∈ ω^{ω}} is countable.

Proof. Let {p_{j}}_{j∈ω} be a one-to-one sequence of all rationals in [0, 1] and let B(p, r) stand for the open ball in [0, 1] with centre p and radius r. Let the set W ⊆ ω^{ω} × [0, 1] be given by

(t, x) ∈ W ⇔ x ∈ ⋃_{j∈ω} B(p_{j}, ∑_{k=j}^{∞} \frac{1}{2^{t(k)+k}}).

Then W is open since it can be written as

W = ⋃_{j∈ω} ⋃_{m=j+1}^{∞} f_{j,m}^{−1}[(−∞, 0)],

where a function f_{j,m} : ω^{ω} × [0, 1] → R given by the formula

f_{j,m}(t, x) = |x − p_{j}| − ∑_{k=j}^{m} \frac{1}{2^{t(k)+k}} ((t, x) ∈ ω^{ω} × [0, 1])

is continuous. Condition (a) clearly follows from the fact that the set

W_{t} = ⋃_{j∈ω} B(p_{j}, ∑_{k=j}^{∞} \frac{1}{2^{t(k)+k}})

is open and dense, for each t ∈ ω^{ω} (thus it is comeager). To check (b), consider any t, t′ ∈ ω^{ω} with t ≤ t′. Then

∑_{k=j}^{∞} \frac{1}{2^{t′(k)+k}} ≤ ∑_{k=j}^{∞} \frac{1}{2^{t(k)+k}} ,

which implies that W_{t′} ⊆ W_{t}. Finally, to obtain (c), let us show that

⋂{W_{t} : t ∈ ω^{ω}} = {p_{j} : j ∈ ω}.

Obviously, we have p_{j} ∈ ⋂{W_{t} : t ∈ ω^{ω}} for every natural number j. Consider any point x ∈ [0, 1] \ {p_{j} : j ∈ ω}. We shall find t ∈ ω^{ω} such that x ∉ W_{t}. First, pick t(0) ∈ ω such that |x − p_{0}| > 1/2^{t(0)}. Suppose now that natural numbers t(0), t(1), …, t(n) are chosen so that

(∗) |x − p_{j}| > ∑_{k=j}^{n} \frac{1}{2^{t(k)+k}} for j = 0, …, n.

Then pick t(n + 1) ∈ ω so that

|x − p_{j}| > ∑_{k=j}^{n} \frac{1}{2^{t(k)+k}} + \frac{1}{2^{t(n+1)+n+1}} for j = 0, …, n,

|x − p_{n+1}| > \frac{1}{2^{t(n+1)+n+1}} .

We have thus defined the sequence t ∈ ω^{ω} satisfying (∗) for each n ∈ ω. If n → ∞ in (∗), we obtain

|x − p_{j}| ≥ ∑_{k=j}^{∞} \frac{1}{2^{t(k)+k}}

for each j ∈ ω. Hence x ∉ W_{t}.

4. Vitali Spaces

In this section, we consider some strong version of the measurability of uncountable unions of measurable sets. Let (E, S, I) be again a measurable space with a σ-ideal. (Recall that two triples connected, respectively, with measure and category are classical examples of a measurable space with a σ-ideal.) We shall say that a family F ⊆ S is countably approximable if there exists a countable subfamily G of F such that

⋃ F \ ⋃ G ∈ I.

Plainly, if a family is countably approximable, then its union is in S but, in general, the converse is not true.

In connection with the problem concerning the S-measurability for unions of uncountable subfamilies of S, it is natural to ask about useful criteria for the countable approximability of such subfamilies provided that the elements of a subfamily have some additional “good” structure. In particular, we are going to consider this question for a triple (E, Br(E), K(E)), where E is a topological space, and for a family of admissible subsets of E. As we know (see Introduction), every admissible set is open modulo NW D(E) and, consequently, belongs to Br(E).

We say that a Vitali covering V of a given subset X of E is admissible if V consists of admissible sets. A space E is called a Vitali space if, for each set X ⊆ E and for each admissible Vitali covering V of X, there exists a disjoint countable family W ⊆ V with X \ ⋃ W belonging to K(E).

We say that E almost satisfies the Suslin condition if there exists a set Z ∈ K(E) such that the subspace E \ Z of E satisfies the Suslin condition. Clearly, if a topological space satisfies the Suslin condition, then it almost satisfies the Suslin condition but it is easy to find examples of spaces which disprove the converse assertion.

We are going to give a full characterization of topological spaces for which any family of admissible sets is countably approximable. For this purpose, we need several auxiliary facts.

We begin with the following consequence of the classical Banach theorem on the unions of open first category sets.

Lemma 4.1. Every topological space E can be expressed in the form E = E_{0} ∪ E_{1} where E_{0} and E_{1} are disjoint sets, E_{0} is an open Baire subspace of E and E_{1} is a closed first category subspace of E.

Proof. (Cf. [8], Theorem 2.4.) According to the Banach Category Theorem (see [1], Theorem 16.1), we have the equality E = E′ ∪ E′′ where E′ is the largest (with respect to inclusion) open meager set in E, and E′′ = E \ E′. Let us put E_{1} = E′ ∪ bd(E′), E_{0} = E \ E_{1}. Since bd(E′) ∈ NW D(E), we have E_{1} ∈ K(E) and, moreover, E_{1} is closed in E. Hence E_{0} is open in E and, by the maximality of E′, we can infer that E_{0} is a Baire space.

The set E_{0} in the above-mentioned expression E = E_{0} ∪E_{1} will be called a Baire kernel of the space E. Obviously, the Baire kernel of E is unique modulo the σ-ideal K(E).

Lemma 4.2. For a topological space E, the following assertions are equivalent:

(1) E almost satisfies the Suslin condition;

(2) the Baire kernel E_{0} of E satisfies the Suslin condition;

(3) for each disjoint family F of open sets in E, there exists a countable family G ⊆ F such that ⋃ F \ ⋃ G ∈ K(E).

The proof of this lemma easily follows from Lemma 4.1 and will be omitted.

Lemma 4.3. Let Z be a subset of a given topological space E. Suppose also that V is an admissible Vitali covering of Z. Then there exists a disjoint family W ⊆ V such that Z \ ⋃ W is a nowhere dense set.

Proof. (Cf. [9], Proposition 1.) We apply Zorn’s lemma to choose W as a maximal (with respect to inclusion) disjoint subfamily of V. To get the assertion, we suppose to the contrary that int(cl(Z \ ⋃ W)) ≠ ∅. Hence we can pick a nonempty open set U ⊆ cl(Z \ ⋃ W). The latter relation implies that Z is dense in U and U ∩ int(W) = ∅ for each W ∈ W. Since all sets from W are admissible, we also have U ∩ W = ∅ for each W ∈ W. Pick x ∈ U ∩ Z. Since V is a Vitali covering of Z, there exists V ∈ V such that x ∈ V ⊆ U. Then W ∪ {V } forms a disjoint subfamily of V which contradicts the maximality of W.

Now, we can formulate and prove

Theorem 4.1. For a topological space E, the following three assertions are equivalent:

(1) E almost satisfies the Suslin condition;

(2) E is a Vitali space;

(3) each family F of admissible subsets of E is countably approximable.

Proof. (1) ⇒ (2). Suppose that (1) is true. Let Z ⊆ E and let V be an admissible Vitali covering of Z. By Lemma 4.3, there exists a disjoint family W ⊆ V such that Z \ ⋃ W ∈ NW D(E). From Lemma 4.2 it follows that there exists a countable family W^{∗} ⊆ W such that ⋃{int(X) : X ∈ W} \ ⋃{int(X) : X ∈ W^{∗}} ∈ K(E). Hence we easily infer that Z\⋃ W^{∗} ∈ K(E). Consequently, E is a Vitali space.

(2) ⇒ (3). Suppose that (2) is true. Let F be a family of admissible subsets of E. We put Z = ⋃{int(X) : X ∈ F}. Then ⋃ F \ Z ∈ NW D(E) (compare with the proof of Theorem 2.1). Let

V = {V : V is open in E & (∃X ∈ F)(V ⊆ int(X))}.

Clearly, V forms an admissible Vitali covering of Z. Since E is a Vitali space, there exists a countable disjoint family W ⊆ V such that Z \ ⋃ W ∈ K(E). For each W ∈ W, choose X_{W} ∈ F such that W ⊆ int(X_{W}) and put G = {X_{W} : W ∈ W}. Then G is a countable subfamily of F and ⋃ F \ ⋃ G ∈ K(E). We have thus proved that the family F is countably approximable.

(3) ⇒ (1). Suppose that (3) is true. Let F be an arbitrary disjoint family of open sets in E. Then F consists of admissible sets. By assumption (3), there exists a countable family G ⊆ F such that ⋃ F \ ⋃ G ∈ K(E). This, by Lemma 4.2, immediately yields assertion (1).

Finally, we wish to note that the last theorem can be generalized (under certain natural assumptions, of course) to the situation of a measurable space with a σ-ideal. This generalization can be obtained by using some abstract versions of the Banach Category Theorem. One of such versions is contained, for instance, in the monograph by Morgan [10].

References

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2. J. Kupka and K. Prikry, The measurability of uncountable unions. Amer. Math. Monthly 91(1984), No. 2, 85–97.

3. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin, 1957.

4. M. Goldstern, An application of Shoenfield’s absoluteness theorem to the theory of uniform distribution. Monatsh. Math. 116(1993), 237–243.

5. K. Kuratowski, Topology, volumes 1,2. Academic Press, New York, 1966, 1968.

6. S. Saks, Theory of the integral. G.E. Stechert, New York, 1937.

7. J. Dieudonné, Foundations of modern analysis. Academic Press, New York, 1960.

8. J. Cichoń, A. Kharazishvili, and B. Wȩglorz, Subsets of the real line, part 1. Lódź University Press, Lódź, 1995.

9. M. Orlando, Vitali covers and category. Exposition. Math. 11(1993), 163–167.

10. J. C. Morgan II, Point set theory. Marcel Dekker, Inc., New York and Basel, 1990.

(Received 09.04.1997)

Authors’ addresses:

Marek Balcerzak

Institute of Mathematics

Lódź Technical University

al. Politechniki 11, 90-924 Lódź

Poland Alexander Kharazishvili Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Georgia