UDC 517.948+513.881

MATHEMATICS

S. N. SLUGIN

AN ANALOG OF THE TOPOLOGY OF A SPACE OF MEASURABLE FUNCTIONS IN A \(K\)-LINEAL WITH UNIT

(Presented by Academician S. L. Sobolev on 17 X 1967)

1. A fundamental system of neighborhoods of zero of a linear topological space \(X\) is called a basis. If the space \(X\) is separated, is a \(K\)-lineal, and has a basis consisting of normal \((^1)\) neighborhoods, then we shall call the basis normal, and \(X\) a \(KT\)-lineal. By convergence of a net \(x_\alpha \to x\) is meant topological convergence. A \(KT\)-lineal has the basic properties of a \(KN\)-lineal \((^1)\). A topologically countably complete \(KT\)-lineal with unit \((b)\) is complete in the sense of \((^{2-4})\) (with respect to the norm of bounded elements).

2. Here an analog of the topology of a space of measurable functions is constructed in a certain foundation \((^1)\) \(Y\) of an Archimedean \(K\)-lineal \(Z\) with unit that is complete in the sense of \((^{2-4})\) (an a.p.e. \(K\)-lineal). Conditions are found under which \(Y=Z\). The basis of the \(KT\)-lineal is constructed essentially in the same way as was done in \((^5)\) for the universal semiring, i.e. an extended \(K\)-space, but under weaker restrictions on the \(K\)-lineal \(Z\) and on the topology of its basis \(I\). The topological continuity of the mapping of a \(KT\)-lineal \(X\), structurally embedded in \(Z\), is established when the topologies coincide in \(X \cap I\).

3. Suppose that in some \(K\)-lineal \(Z\) there is distinguished a certain system of normal sets \(W\), satisfying the first axiom of separatedness at zero: \(\cap W=\{0\}\), and the requirements imposed on a basis of a linear topological space (see, for example, Theorem 1 (1.XI) in \((^6)\)), except for one: some of the sets \(W\) may not absorb any elements in \(Z\). Then \(Z\) shall be called a generalized \(KT\)-lineal with basis \(\{W\}\).

If some normal sublineal \(Y\) of the generalized \(KT\)-lineal \(Z\) with unit and basis \(\{W\}\) contains its unit and is a \(KT\)-lineal with basis \(\{W \cap Y\}\), then \(Y\) shall be called a \(KT\)-foundation in \(Z\).

If each set \(W\) absorbs the unit of the generalized \(KT\)-lineal \(Z\), then the sublineal \(Z_0\) of bounded elements is a \(KT\)-foundation in \(Z\). It is clear that any \(KT\)-foundation \(Y \supset Z_0\).

If each set \(W\), moreover, absorbs any element of a certain linear substructure \(X\) of the \(K\)-lineal \(Z\), then there exists a \(KT\)-foundation \(Y \supset X+Z_0\).

1. An a.p.e. \(K\)-lineal (see above, item 2) \(Z\) is realized \((^{3,4})\) as the \(K\)-lineal \(Z(Q)\) of certain continuous functions, allowed to take infinite values, on a certain bicompact \((^1)\) \(Q\). Within \(Z\) there is the \(K\)-lineal \(Z_0=C(Q)\) of all continuous bounded functions \((^2)\). The base \(I\) of unit elements \(e\) is structurally isomorphic to the system of open-closed sets \(E_e \subset Q\).

Suppose there is a system of subsets \(\Gamma \subset I\) satisfying the conditions:

\[ (\Gamma 1)\quad \text{For any } \Gamma_i \text{ there exists } \Gamma \subset \Gamma_1 \cap \Gamma_2. \]

\[ (\Gamma 2)\quad \text{For each } \Gamma_0 \text{ there is } \Gamma \text{ such that if } e_i \in \Gamma,\text{ then } e_1 \vee e_2 \in \Gamma_0. \]

\[ (\Gamma 3)\quad \text{All } \Gamma \text{ are normally contained in } I. \]

\[ (\Gamma 4)\quad \cap \Gamma=\{0\}. \]

Define in \(Z\) the sets \(W_{\Gamma\varepsilon}\) \((^5)\): \(z \in W_{\Gamma\varepsilon}\) if there exists an element \(e \in \Gamma\) such that \(\{t;\ |z(t)| \geq \varepsilon\} \subset E_e\) (here \(0 < \varepsilon \leq 1\)).

The following assertions hold.

a) In \(Z\) there is a \(KT\)-fundament \(Y\) with basis \(\{Y \cap W_{\Gamma\varepsilon}\}\), for example \(Y=Z_0\).

b) The system \(\{\Gamma\}\) determines a topology of the basis \(I\),

\[ \Gamma = I \cap W_{\Gamma\varepsilon}. \]

The basis \(I\) is a (topological) subspace of the \(KT\)-lineal \(Y\). Topological convergence of a direction \(e_\alpha \to e\) in \(I\) means that the symmetric difference \(e_\alpha C e + e C e_\alpha \in \Gamma\) for \(\alpha \geq \alpha_0\).

c) For any finite set of elements \(y_i\) of the \(KT\)-fundament \(Y\) and an arbitrary neighborhood \(\Gamma\) there exists an element \(e \in \Gamma\) such that

\[ \bigcup_i \{t;\ y_i(t)=\infty\} \subset E_e. \]

On the set \(E_{Ce}=Q \setminus E_e\) the functions \(y_i(t)\) are bounded.

d) In the \(KT\)-fundament \(Y\) with basis \(\{Y \cap W_{\Gamma\varepsilon}\}\), an abstract function generated \((^4)\) in an a.p.e. \(K\)-lineal \(Z\) by a real continuous function of several variables is topologically continuous at all points in whose vicinity the abstract function has meaning in \(Y\).

e) In order that a certain normal sublineal \(Y\), containing the unit of the given \(K\)-lineal \(Z\), be a \(KT\)-fundament with basis \(\{Y \cap W_{\Gamma\varepsilon}\}\), it is necessary and sufficient that the following condition be satisfied:

e′) for every element \(y \in Y\) and arbitrary neighborhood \(\Gamma\) there is an element \(e \in \Gamma\) such that

\[ \{t;\ y(t)=\infty\} \subset E_e. \]

1. Let now the bicompactum \(Q\) be completely disconnected \((^1)\). A special case of an a.p.e. \(K\)-lineal \(Z(Q)\) with a completely disconnected bicompactum \(Q\) is a \(K_\sigma\)-space \((^1)\) with a unit.

f) For every element \(z \in Z\) and arbitrary quantities \(\mu,\nu:\ \mu < \nu \leq +\infty\), there exists an element \(e\) such that

\[ \{t;\ z(t) \geq \nu\} \subset E_e,\qquad z \geq \mu e. \]

g) Condition e′) is equivalent to the following relation between monotone \((r)\)-convergence of a sequence of unit elements and topological convergence:

\[ (\Gamma5)\quad \text{If } e_n \geq e_{n+1} \xrightarrow{(r)} 0 \text{ in } Y,\text{ then } e_n \to 0 \text{ in } I. \]

If condition \((\Gamma5)\) is satisfied for \(Y=Z\), then \(Z\) is a \(KT\)-lineal with basis \(\{W_{\Gamma\varepsilon}\}\).

h) In order that the whole \(K\)-lineal \(Z\) be a \(KT\)-lineal with basis \(\{W_{\Gamma\varepsilon}\}\), it is sufficient that monotone \((o)\)-convergence of a sequence in \(I\) imply topological convergence:

\[ (A1)\quad \text{If } e_n \downarrow 0,\text{ then } e_n \to 0. \]

Condition (A1) is weaker than that indicated in axiom 6 on p. 52 in \((^5)\).

The topology defined by the basis \(\{W_{\Gamma\varepsilon}\}\) or by its trace \(\{Y \cap W_{\Gamma\varepsilon}\}\) is an analogue of the topology of the space of measurable functions; the sets \(E_e\) play the role of measurable sets in the realization of \(K B\)-spaces \((^{7-10})\).

i) If from the convergence \(e_\alpha \xrightarrow{(0)} 0\) \((\alpha \in A)\) it follows that \(e_\alpha \to 0\), then from the convergence \(y_\alpha \xrightarrow{(0)} y\) \((\alpha \in A)\) it follows that \(y_\alpha \to y\).

1. Let a certain \(KT\)-lineal \(X\) with normal basis \(\{V\}\) be a normal sublineal of some a.p.e. \(K\)-lineal \(Z\). Put

\[ \Gamma = V \cap I. \tag{1} \]

Then all conditions \((\Gamma1—4)\) are fulfilled.

If the bicompactum \(Q\) is totally disconnected, then

\[ \mu V \subset W_{\Gamma \varepsilon}\qquad (0<\mu<\varepsilon); \]

in \(Z\) there is a \(KT\)-fundament \(Y \supset X+Z_0\); the \(KT\)-lineal \(X\) with basis \(\{V\}\) is topologically continuously embedded in the \(KT\)-lineal \(Y\) with basis \(\{Y\cap W_{\Gamma \varepsilon}\}\).

1. Let some \(KT\)-lineal \(X\) with unit and normal basis \(\{V\}\) be a normal sublineal of some inner-normal \((^{4})\) a.p. \(K\)-lineal \(Z(Q)\) with the same unit (a particular case of such a \(K\)-lineal \(Z\) is a \(K_\sigma\)-space with unit), and let the neighborhoods \(\Gamma\) be defined by equality (1). Then, in order that the mapping of the \(KT\)-lineal \(X\) with basis \(\{X\cap W_{\Gamma \varepsilon}\}\) into the \(KT\)-lineal \(X\) with basis \(\{V\}\) be topologically continuous, it is necessary and sufficient that the condition be fulfilled (see axiom 8 on p. 52 in \((^{5})\)):

\((\Gamma 6)\) For every neighborhood \(V_0\) there exists a neighborhood \(\Gamma\) such that the algebraic product \(\Gamma X\subset V_0\).

If, moreover, the bicompactum \(Q\) is totally disconnected, then the indicated mapping is topologically bicontinuous.

Gorky State University
named after N. I. Lobachevsky

Received
11 IX 1967

CITED LITERATURE

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