Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ON LEBESGUE THEOREM FOR MULTIVALUED FUNCTIONS OF TWO VARIABLES

GRAŻYNA KWIECIŃSKA

Abstract. In the paper we investigate Borel classes of multivalued functions of two variables. In particular we generalize a result of Marczewski and Ryll-Nardzewski [6] concerning of real function whose ones of its sections are right-continuous and other ones are of Borel class α, into the case of multivalued functions.

1. Introduction

Many results were publisched about the Borel classification of multivalued functions depending on the one variable (see [5, 3, 1, 4, 7, 8]). In the case of multivalued function of two variables we have the possibility of formulation of hypotheses concerning of its sectionwise properties.

Lebesgue has shown that any real function f of two variables with continuous ones of its sections and of Borel class α the other ones is of Borel class α + 1. Marczewski and Ryll-Nardzewski have shown (see [6]) that the condition of continuity in this theorem may be replaced by right-continuity (or left-continuity). In this paper we generalize these results into the case of multivalued functions in possible general abstract spaces.

2. Preliminaries

Let T and Z be two nonempty sets and let Φ : T → Z be a multivalued function, i.e. Φ denotes a mapping such that Φ(t) is a nonempty subset of Z for t ∈ T. Then two inverse images of a subset G ⊂ Z may be defined:

Φ^{+}(G) = {t ∈ T : Φ(t) ⊂ G}

and

Φ^{−}(G) = {t ∈ T : Φ(t) ∩ G ≠ ∅}.

2000 Mathematics Subject Classification. 54C60, 54C08, 28B20.

Key words and phrases. multivalued functions, semi-continuity of multivalued functions, Baire classes of multivalued functions.

This research was supported by the University of Gdańsk, grant BW Nr 5100-5-0188-9.

Grażyna Kwiecińska, On Lebesgue Theorem for multivalued functions of two variables, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 181–189, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204133.

The following relations hold betwen these inverse images:

(1)  Φ^{−}(G) = T \ Φ^{+}(Z \ G) and Φ^{+}(G) = T \ Φ^{−}(Z \ G).

Let (T, T (T)) and (Z, T (Z)) be topological spaces. The notations Int(A) and Cl(A) will be used to denote, respectively, the interior and the closure of a set A.

Definition 1. A multivalued function Φ : T → Z is said to be T (T)-upper (resp. T (T)-lower) semicontinuous at a point t ∈ T if

∀G ∈ T (Z) (Φ(t) ⊂ G ⇒ t ∈ IntΦ^{+}(G))

(resp. ∀G ∈ T (Z) (Φ(t) ∩ G ≠ ∅ ⇒ t ∈ IntΦ^{−}(G))).

F is called T (T)-continuous at the point t if it is simultaneously T (T)- upper and T (T)-lower semicontinuous at t.

A multivalued function Φ being T (T)-upper (resp. T (T)-lower) semicontinuous at each point t ∈ T is said to be T (T)-upper (resp. T (T)-lower) semicontinuous.

It is clear that a multivalued function Φ is T (T)-upper (resp. T (T)-lower) semicontinuous if and only if Φ^{+}(G) ∈ T (T) (resp. Φ^{−}(G) ∈ T (T)), whenever G ∈ T (Z).

Given any countable ordinal number α, let ∑_{α}(T) and Π_{α}(T) denote the additive and multiplicative class α, respectively, in the Borel hierarchy of subsets of the topological space (T, T (T)).

We shall always assume α to be an arbitrary countable ordinal number. In perfect spaces the following inclusions hold:

(2)  ∑_{α}(T) ⊂ Π_{α+1}(T) ⊂ ∑_{α+1}(T).

Definition 2. A multivalued function Φ : T → Z will be said to be of T (T)-lower (resp. T (T)-upper) Borel class α if

Φ^{−}(G) ∈ ∑_{α}(T)

(resp. Φ^{+}(G) ∈ ∑_{α}(T)), whenever G ∈ T (Z).

Let us note that a multivalued function of T (T)-lower (resp. T (T)-upper) class 0 is T (T)-lower (resp. T (T)-upper) semicontinuous.

Let f : T → ℝ and g : T → ℝ be point-valued functions. Then a multivalued function Φ : T → ℝ defined by formula

(3)  Φ(t) = [f(t), g(t)] ⊂ ℝ

is of T (T)-lower (resp. T (T)-upper) Borel class α if and only if f is of T (T)- upper (resp. T (T)-lower) and g is of T (T)-lower (resp. T (T)-upper) class α in the Young classification.

In fact, for a < b we have

Φ^{−}((a, b)) = {t ∈ T : f(t) < b} ∩ {t ∈ T : g(t) > a}

and

Φ^{+}((a, b)) = {t ∈ T : f(t) > a} ∩ {t ∈ T : g(t) < b}.

3. Main results

Let F : X × Y → Z be a multivalued function and (x_{0}, y_{0}) ∈ X × Y. Then a multivalued function F_{x_{0}} : Y → Z such that F_{x_{0}}(y) = F (x_{0}, y) is called x_{0}-section of F. Similarly a multivalued function F^{y_{0}} : X → Z such that F^{y_{0}}(x) = F (x, y_{0}) is called y_{0}-section of F.

Theorem 1. Let (Y, d) be a metric space and (X, T (X)), (Z, T (Z)) two perfectly normal topological spaces. Let T (Y) be a topology on Y which is finer than the metric one and such that (Y, T (Y)) is separable. Let S be a countable T (Y)-dense subset of Y. Suppose that to every point v ∈ Y there corresponds a subset U(v) ∈ T (Y) such that

∀y ∈ S B(y) = {v : y ∈ U(v)} ∈ ∑_{α}(Y, d)

and

∀v ∈ Y N (v) = {U(v) ∩ B(v, 2^{−n}) : n = 1, 2, …},

where B(v, 2^{−n}) denotes the open ball centered in v with radius 2^{−n}, forms a filterbase of T (Y)-neighbourghoods of the point v.

Assume that F : X ×Y → Z is a multivalued function whose all y-sections are of upper class α and all x-sections are T (Y)-continuous. Then F is of lower class α + 1 on the product (X, T (X)) ⊗ (Y, d).

Proof. Let D be an arbitrary T (Z)-closed subset of Z. By (1) it is enough to show that

F^{+}(D) ∈ ∏_{α+1}((X, T (X)) ⊗ (Y, d)).

Since Z is perfectly normal, there is a sequence {G_{n}}_{n∈ℕ} of T (Z)-open sets such that

(4)  D = ⋂_{n∈ℕ}G_{n} = ⋂_{n∈ℕ}Cl(G_{n})

and

(5)  Cl(G_{n+1}) ⊂ G_{n} for n ∈ ℕ.

Let S = {y_{k} : k ∈ ℕ}. We will prove that

(6)  F^{+}(D) = ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ⊂ G_{n}} × V_{n}(y_{k})),

where

(7)  V_{n}(y_{k}) = {v ∈ Y : y_{k} ∈ U(v)} ∩ B(v, 2^{−n}).

Let

(u, v) ∈ F^{+}(D) = {(x, y) ∈ X × Y : F (x, y) ⊂ D}.

Then F (u, v) ⊂ G_{n} for each n ∈ ℕ, by (4). Let n be fixed. By the T (Y)- upper semicontinuity of the u-section of F at the point v ∈ Y there is a T (Y)-open neighbourhood U(v) ∈ N (v) of v such that F (u, y) ⊂ G_{n} for any y ∈ U(v).

Let

K = {m ∈ ℕ : y_{m} ∈ U(v) ∩ S}

and let

k = min{m ∈ K : v ∈ V_{n}(y_{m})}.

Then

(u, v) ∈ [F^{y_{k}}]^{+}(G_{n}) × V_{n}(y_{k})

and the inclusion

(8)  F^{+}(D) ⊂ ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ⊂ G_{n}} × V_{n}(y_{k}))

is proved.

Conversely, let (u, v) belongs to the right-hand side of (6). Suppose that (u, v) ∉ F^{+}(D). Then by (4) we must have

(9)  F (u, v) ∩ (Z \ Cl(G_{m}) ≠ ∅ for some m ∈ ℕ.

By T (Y)-lower semicontinuity of the u-section of F at the point v ∈ Y there is a T (Y)-open neighbourhood W (v) ∈ N (v) of v such that

(10)  F (u, y) ∩ (Z \ Cl(G_{m}) ≠ ∅ for any y ∈ W (v).

We have supposed that

(u, v) ∈ ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ⊂ G_{n}} × V_{n}(y_{k})).

Therefore we conclude from (4) that to each n there corresponds an index k = k(n) such that

(11)  F (u, y_{k(n)}) ⊂ G_{n}.

For v ∈ V_{n}(y_{k(n)}) ⊂ B(v, 2^{−n}) we obtain lim_{n→∞} d(v, y_{k(n)}) = 0. Since y_{k(n)} tends to v in (Y, d) as n tends to infinity, (10) and (11) show that there is an index n_{0} such that

(12)  F (u, y_{k(n)}) ∩ (Z \ Cl(G_{m})) ≠ ∅ for any n > n_{0}.

By (5) and (11) we have

F (u, y_{k(n)}) ⊂ G_{n} ⊂ G_{n−1} ⊂ …

for n ∈ ℕ.

In particular,

F (u, y_{k(n+j)}) ⊂ G_{n+j} ⊂ G_{n}

for any j ∈ ℕ. Fixing now n = m (see (9)) we obtain F (u, y_{k(m+j)} ⊂ G_{m} for any j ∈ ℕ, which contradicts (12). We must have

∃n ∈ ℕ ∀y ∈ S v ∉ V_{n}(y) ∨ F (u, y) ⊄ G_{n}.

This formula means that

(u, v) ∉ ⋂_{n∈ℕ} ⋃_{k∈ℕ}([F^{y_{k}}]^{+}(G_{n}) × V_{n}(y_{k}))

and the inclusion

(13)  ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ⊂ G_{n}} × V_{n}(y_{k})) ⊂ F^{+}(D)

holds. By (8) and (13) the equality (6) is proved.

Observe that

{x : F (x, y_{k}) ⊂ G_{n}} ∈ ∑_{α}(X, T (X))

since y_{k}-section of F is of upper class α. Furthermore it is assumed that V_{n}(y_{k}) ∈ ∑_{α}(Y, d). Therefore by (6) F^{+}(D) is a countable intersection of countable unions of the sets of the class

∑_{α}(X, T (X)) ⊗ ∑_{α}(Y, d) ⊂ ∑_{α}(X × Y),

where X ×Y is the product of topological spaces (X, T (X)) and (Y, d). This completes the proof of Theorem 1. □

We give below two examples of topology T (Y) on Y fulfilling requirements of Theorem 1. From these examples it will be clear, that the x-sections of a multivalued function F in Theorem 1 may be either all right-continuous or all left-continuous in some meaning.

Example 1. Let (Y, ⋄, d) be a topological group, whose topology is induced by an invariant distance function d (i.e. d(θ, y) = d(v, y⋄v)), where θ denotes a neutral element of Y. Assume furthermore that (Y, d) is separable.

Let U ⊂ Y be an open set such that θ is an accumulation point of U. Let

U_{n} = (B(θ, 2^{−n}) ∩ U) ∪ {θ} and V_{n}(y) = y ⋄ U_{n} = {y ⋄ v : v ∈ U_{n}}

for n ∈ ℕ. Then {V_{n}(y)}_{n∈ℕ} forms a filterbase of neighbourhoods of a point y ∈ Y and the topology T (Y) in Y generated by this base fulfils all requirements of Theorem 1.

Indeed, it suffices to prove that {U_{n}}_{n∈ℕ} forms a base of neighborhoods of θ. We have

U_{n} ∩ U_{m} = U_{min(n,m)}.

Let n ∈ ℕ and v ∈ U_{n}. Then there is k ∈ ℕ such that

B(v, 2^{−k}) = v ⋄ B(θ, 2^{−k}) ⊂ U_{n}.

Therefore

∀n ∈ ℕ ∀v ∈ U_{n} ∃k ∈ ℕ V_{k}(v) ⊂ U_{n}.

A countable dense subset of (Y, d) is also T (Y)-dense. It remains to show that V_{n}(y) is a Borel set in (Y, d) for any n ∈ ℕ. Let n ∈ ℕ and let Φ : Y → Y be a multivalued function defined by formula Φ(y) = V_{n}(y). Then Φ is continuous and and its graph

Gr(Φ) = {(y, v) : v ∈ Φ(y)}

is homeomorphic to the set

Y × U_{n} ∈ ∑_{1}(Y, d) ⊗ (Y, d) ∩ ∏_{1}(Y, d) ⊗ (Y, d).

Finally V_{n}(y) ∈ ∑_{1}(Y, d) ∩ ∏_{1}(Y, d) for each n ∈ ℕ.

Example 2. Let (Y, d, ≤) be a linearly ordered metric space. We follow Dravecky and Neubrunn (see [2]) in assuming that the space (Y, d, ≤) has the property U, i.e. (Y, ≤) is linearly ordered and there is a countable dense set S in (Y, d,) such that for any y ∈ Y we have y = lim_{n→∞} y_{n}, where y_{n} ∈ S and y ≤ y_{n} for n ∈ ℕ. Then the topology T (Y) on Y generated by all open sets in (Y, d) and also by all intervals I_{a} = {y ∈ Y : y ≤ a}, a ∈ Y, fulfills the assumptions of Theorem 1. Indeed, let y ∈ Y and r > 0. Then

U_{r}(y) = B(y, r) ∩ I_{y} = {x ∈ Y : d(x, y) < r ∧ x ≤ y}

is a T (Y)-neighbourhood of the point y.

Let x ∈ U_{r}(y). Then x ∈ B(y, r) and x ≤ y, and then there is r_{1} > 0 such that d(x, y) = r − r_{1}. Let δ < min(r, r_{1}). Then B(x, δ) ⊂ B(y, r). Let n ∈ ℕ be such a number that 2^{−n} < δ. Then U_{2^{−n}}(x) ⊂ U_{r}(y) and we see that {U_{2^{−n}}(y)}_{n∈ℕ} forms a filterbase of T (Y)-neighbourhoods of the point y.

The set S is also T (Y)-dense. It remains to show that the set

V_{r}(y) = {z ∈ Y : y ∈ U_{r}(Z)}

is a Borel set in (Y, d). First we will show that

(14)

If y_{0} ≠ y and y_{0} ∈ V_{r}(y), then there exists 0 < r_{1} < r
such that U_{r_{1}}(y_{0}) ⊂ V_{r}(y)

Suppose, contrary to our claim, that U_{r_{1}}(y_{0}) ⊄ V_{r}(y) for any r_{1} < r. Now let n ∈ ℕ be such that \frac{1}{n} < r. Then there is y_{n} such that y ≤ y_{n} and y_{n} ∈ U_{\frac{1}{n}} (y_{0}) \ V_{r}(y), and then

y ≤ y_{n} ∧ d(y_{n}, y_{0}) < \frac{1}{n} ∧ y_{n} ≤ y_{0} ∧ (y_{n} ≤ y ∨ d(y_{n}, y) ≥ r)

for n > \frac{1}{n}. If it were true that d(y_{n}, y_{0}) < \frac{1}{n} and y ≤ y_{n} ≤ y_{0} and y_{n} ≤ y, we would have

lim_{n→∞} y_{n} = y_{0} = y,

in contradiction with y ≠ y_{0}. Let d(y_{0}, y) = ε. If it were true that d(y_{n}, y_{0}) < \frac{1}{n} and d(y_{n}, y) ≥ r we would have

r ≤ d(y_{n}, y) ≤ d(y_{n}, y_{0}) + d(y_{0}, y) < \frac{1}{n} + ε.

Then we would have \frac{1}{n} > r − ε > 0 for almost every n ∈ ℕ, which is impossible. This establishes (14).

Our next claim is that

(15)

If y_{0} ≠ y and y_{0} ∈ V_{r}(y), then there is δ > 0
such that B(y_{0}, δ) ⊂ V_{r}(y).

Indeed, according to (14) there is r_{1} ∈ (0, r) such that U_{r_{1}}(y_{0}) ⊂ V_{r}(y). Let ε = d(y_{0}, y) < r and let δ < min(ε, r − ε, r_{1}). Let z ∈ B(y_{0}, δ). Then either d(y_{0}, z) < δ and z ≤ y_{0} or d(y_{0}, z) < δ and y_{0} ≤ z. In the first case z ∈ U_{δ}(y_{0}) ⊂ V_{r}(y). In the second one

d(z, y) ≤ d(z, y_{0}) + d(y_{0}, y) < δ + ε < r − ε + ε = r

and y ≤ z show that z ∈ V_{r}(y). Combining these both results we conclude that B(y_{0}, δ) ⊂ V_{r}(y) and (15) is proved.

By (15) we see that the set

{z ∈ Y : d(z, y) < r ∧ y ≤ z ∧ y ≠ z}

is open in (Y, d). Therefore

V_{r}(y) = {y} ∪ {z ∈ Y : d(z, y) < r ∧ y ≤ z ∧ y ≠ z} ∈ ∑_{1}(Y, d) ∩ ∏_{1}(Y, d).

Note that this topology T (Y) may be viewed as a natural generalization of the known Sorgenfrey topology on the real line.

Corollary 1. Let f be a real function defined on the product of perfectly normal toplogical space X and the real line ℝ. Let us suppose that all xsections of f are right-continuous and all y-sections of f are of upper Young class α. Then f is of lower class α + 1 on X × ℝ, i.e. it may be represented as a point-limit of an increasing sequence of functions of upper Young class α.

Proof. Let us note that a multivalued function F : X × ℝ → ℝ defined by formula

F (x, y) = [2 − arctan f(x, y), 2 + arctan f(x, y)]

is of lower class α + 1, by Theorem 1. Moreover for a < b we have

F^{−}((a, b)) = {(x, y) : 2−arctan f(x, y) < b}∩{(x, y) : 2+arctan f(x, y) > a}.

By (3) the function g(x, y) = 2 − arctan f(x, y) is of upper class α + 1 and the function h(x, y) = 2 + arctan f(x, y) is of lower class α + 1 in the Young classification, which finishes the proof of Corollary 1. □

The next theorem is a dualization of Theorem 1.

Theorem 2. Let (Y, d) be a metric space and (X, T (X)), (Z, T (Z)) two perfectly normal topological spaces. Let T (Y) be a topology on Y which is finer than the metric one and such that (Y, T (Y)) is seperable. Let S be a countable T (Y)-dense subset of Y. Suppose that to every point v ∈ Y there corresponds a subset U(v) ∈ T (Y) such that

∀y ∈ S B(y) = {v : y ∈ U(v)} ∈ ∑_{α}(Y, d)

and

∀v ∈ Y N (v) = {U(v) ∩ B(v, 2^{−n}) : n = 1, 2, …},

forms a filterbase of T (Y)-neighbourghoods of the point v. Let F : X × Y → Z be a compact-valued multivalued function whose all y-sections are of lower class α and all x-sections are T (Y)-continuous. Then F is of upper class α + 1 on the product (X, T (X)) ⊗ (Y, d).

Proof. Let D be an arbitrary T (Z)-closed subset of Z and let S = {y_{k} : k ∈ ℕ. We will first prove that

(16)  F^{−}(D) = ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ∩ G_{n} ≠ ∅} × V_{n}(y_{k})),

where G_{n} are open subsets of Z fulfilling (4) and (5), while V_{n}(y_{k}) is defined by the formula (7).

If

(u, v) ∈ F^{−}(D) = {(x, y) : F (x, y) ∩ D ≠ ∅},

then by (4) F (u, v) has nonempty intersection with G_{n} for each n ∈ ℕ. Let n be fixed and arbitrary. By T (Y)-lower semicontinuity of u-section of F at the point v there exists a T (Y)-open neighbourhood U(v) ∈ N (v) of v such that F (u, y) ∩ G_{n} ≠ ∅ for all y ∈ U(v). Taking k such that v ∈ V_{n}(y_{k}) we have

(u, v) ∈ [F^{y_{k}}]^{−}(G_{n}) × V_{n}(y_{k}) = {x : F (x, y_{k}) ∩ G_{n} ≠ ∅} × V_{n}(y_{k}),

which gives

F^{−}(D) ⊂ ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ∩ G_{n} ≠ ∅} × V_{n}(y_{k})).

Now let us suppose that

(u, v) ∈ ⋂_{n∈ℕ} ⋃_{k∈ℕ}({x : F (x, y_{k}) ∩ G_{n} ≠ ∅} × V_{n}(y_{k})).

Then to each n there corresponds an index k = k(n) such that for y_{k(n)} ∈ S we have F (u, y_{k(n)}) ∩ G_{n} ≠ ∅, and then by (5)

(17)  F (u, y_{k(n+j)}) ∩ G_{n} ≠ ∅ for any j ∈ ℕ.

If (u, v) were not in F^{−}(D), by (4) we would have

F (u, v) ⊂ Z \ D = ⋃_{n∈ℕ}(Z \ Cl(G_{n})).

The value F (u, v) is a compact subset of Z and the sets Z \ Cl(G_{n}), n ∈ ℕ, create a decreasing sequence of open sets, i.e.

Z \ Cl(G_{n}) ⊂ Z \ Cl(G_{n+1}).

Therefore for some m ∈ ℕ we have F (u, v) ⊂ Z \ Cl(G_{m}). Then by the T (Y)-upper semicontinuity of u-section of F at the point v ∈ Y we have F (u, y) ⊂ Z \ Cl(G_{m}) for y ∈ W (v), where W (v) is a certain neighbourhood of the point v, chosen from the postulated filterbase N (v). Since y_{k(n)} tends in (Y, d) to v as n tends to infinity, by the above there exists an index n_{0} such that y_{k(n)} ∈ W (v) for n > n_{0}. Therefore

(18)  F (u, y_{k(n)}) ⊂ Z \ Cl(G_{m}) for any n > n_{0}.

Taking n = m in (17) we have F (u, y_{k(m+j)}) ∩ G_{m} ≠ ∅ for any j ∈ ℕ, which contradicts (18). Thus the equality (16) is proved.

Since the y_{k}-section of F is of lower class α, we have

{x : F (x, y_{k}) ∩ G_{n} ≠ ∅} ∈ ∑_{α}(X).

Moreover under the assumption of our theorem we have V_{n}(y_{k}) ∈ ∑_{α}(Y, d). Thus we conclude from (16) that

F^{−}(D) ∈ ∑_{α}(X) ⊗ ∑_{α}(Y, d) ⊂ ∑_{α}(X ⊗ Y) ⊂ ∏_{α+1}(X ⊗ Y),

where X ⊗ Y is the product of topological spaces (X, T (X)) and (Y, d), as required. The proof of Theorem 2 is finished. □

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University of Gdańsk, Institute of Mathematics, Wita Stwosza 57, 80-952 Gdańsk, Poland

   E-mail address: gkk@math.univ.gda.pl