UDC 517.397:519.53:513.88

MATHEMATICS

L. Ya. SAVEL'EV

ON VECTOR COUNTABLY ADDITIVE MEASURES

(Presented by Academician L. V. Kantorovich, 3 VII 1965)

1. Introduction. Consider: 1) the class \(\mathscr P\) of all subsets of a set \(U\) with the usual ring structure, order, and \(so\)-topology \(S\) ((\(^{1}\), § 2); 2) a ring \(\mathscr R\) in \(\mathscr P\); 3) the system \(\mathbf K=\mathbf K(\mathscr R)\) \((\mathbf D=\mathbf D(\mathscr R))\) of all finite (countable) disjoint parts of \(\mathscr R\) \((\mathbf K\subseteq \mathbf D)\); 4) a complete separable real locally convex space \(G\).

Definition 1. We shall call a mapping \(a\) of the ring \(\mathscr R\) into \(G\) countably additive in \(\mathscr R\) if, for every \(\mathscr D\in \mathbf D\) for which \(\sum \mathscr D\in \mathscr R\), the family \(a\mathscr D\) is summable and \(\sum a\mathscr D=a\sum \mathscr D\).

Denote by \(A=A(\mathscr R,G)\) the set of all mappings of \(\mathscr R\) into \(G\) that are countably additive in \(\mathscr R\).

Definition 2. We shall call a mapping \(a\in A\) countably additive if, for every \(\mathscr D\in \mathbf D\), the family \(a\mathscr D\) is summable.

Denote by \(C=C(\mathscr R,G)\) the set of all \(a\in A\) that are countably additive mappings.

Denote by \(\Gamma\) the set of seminorms on \(G\) defining the topology in \(G\). For each \(\gamma\in \Gamma\), \(a\in A\), \(X\in \mathscr R\), define
\[ v_{\gamma a}X=\sup\{\sum \gamma a\mathscr K:\mathscr K\in \mathbf K,\ \sum \mathscr K\subseteq X\}; \]
we shall call the mappings \(v_{\gamma a}:X\to v_{\gamma a}X\) the variations for \(a\).

Denote by \(V=V(\mathscr R,G)\) the set of all \(a\in A\) having bounded variations; \(S=S(\mathscr R,G)\)—those that are continuous mappings of \(\mathscr R\) into \(G\); \(B=B(\mathscr R,G)\)—those that are bounded mappings of \(\mathscr R\) into \(G\); \(E=E(\mathscr R,G)\)—those having an extension \(\bar a\in A(\bar{\mathscr R},G)\) (\(\bar{\mathscr R}\) is the closure of the ring \(\mathscr R\) in \(\mathscr P\)).

The note is devoted to describing the connection between the distinguished classes of countably additive mappings. The main result is expressed as follows:

Theorem 1. \(V\subseteq S(E)\subseteq C\subseteq B\).

2. The topology \(V_a\). For each \(a\in V\), \(\gamma\in \Gamma\), \(K\in \mathscr K\) (\(\mathscr K\) is the class of all finite subsets of the set \(\Gamma\)), \(\varepsilon>0\), define
\[ \mathfrak U(\gamma,\varepsilon)=\{X\in \mathscr P:X\subseteq \sum\mathscr D_0\ (\mathscr D_0\in \mathbf D),\ \sum v_{\gamma a}\mathscr D_0<\varepsilon\}; \quad \mathfrak U(K,\varepsilon)=\bigcap_{\gamma\in K}\mathfrak U(\gamma,\varepsilon). \]
For every \(\gamma\in \Gamma\), \(a\in V\), the variation \(v_{\gamma a}\in B(\mathscr R,R)\). Consequently, there exists a topology \(V_{\gamma a}\) in \(\mathscr P\), compatible with the ring structure in \(\mathscr P\), for which \(\{\mathfrak U(\gamma,\varepsilon)\}_\varepsilon\) is a fundamental system of neighborhoods of zero ((\(^{2}\), § 1). Consider the topology \(V_a\) in \(\mathscr P\) generated by the union \(\bigcup (V_{\gamma a})_\gamma\). The following propositions describe some general properties of \(V_a\).

Proposition 1. The topology \(V_a\) is compatible with the ring structure in \(\mathscr P\), and \(\{\mathfrak U(K,\varepsilon)\}_{K,\varepsilon}\) is a fundamental system of neighborhoods of zero.

Proposition 2. Every class \(\mathscr D\in \mathbf D(\bar{\mathscr R})\) is summable, and its sum is equal to the union \(\sum \mathscr D\).

Denote by \(\bar{\mathscr R}_a\) the closure of the ring \(\mathscr R\) in the topology \(V_a\).

Proposition 3. \(\bar{\mathscr R}\subseteq \bar{\mathscr R}_a\).

Proposition 4. Every \(a\in V\) is uniformly continuous \((V_a)\).

3. Scheme of the proof of Theorem 1. Theorem 1 is the union of the following propositions.

Proposition 5. \(V\subseteq S\).

For every \(\gamma\in \Gamma\), \(a\in V\), \(v_{\gamma a}\in B(\mathscr R,R)\). Hence \(v_{\gamma a}\in S(\mathscr R,R)\) ((\(^{1}\), § 3, Theorem 3). Hence the continuity of \(a\) in \(O(S)\) and the continuity of \(a\).

Proposition 6. \(S \subseteq C\).

For each \(a \in C'\), inductively, using the Cauchy criterion \(\bigl((^3),\) Ch. 3, § 4, item 2, Theorem 1\()\), a sequence \((D_n)\) is defined such that \(\{D_n\}\in \mathbf D\) and \((aD_n)\to 0\).

Proposition 7. \(V \subseteq E\).

For each \(a \in V\) there exists a uniformly continuous extension \(\bar a\) (Proposition 4; \((^3)\), Ch. 2, § 3, item 4, Theorem 1). \(\bar a\) is finitely additive (cf. \((^2)\), § 2, Lemma 5). For each \(\mathcal D \in \mathbf D(\mathcal R)\), by Proposition 2, \(\sum \mathcal D \subseteq \bar{\mathcal F}_a\). Consequently, \(\bar a \in \mathbf A(\mathcal R_a,G)\) and \(a\in E\) (Proposition 3).

Proposition 8. \(E \subseteq C\).

This follows directly from the definitions.

Proposition 9. \(C \subseteq B\).

This follows from the particular case for numerical mappings; Proposition 5, item 5, § 4, Ch. 3 \((^3)\) and the corollary of Theorem 3, item 4, § 2, Ch. 4 \((^4)\).

4. Some corollaries

Corollary 1. If \(G\) is finite-dimensional, then \(V=E=S=C=B\).

It is enough to prove that \(B\subseteq V\) and to consider the case \(G=R^n\) (Theorem 1; \((^4)\), Ch. 1, § 2, item 3, Theorem 2). The proof reduces to estimating the variation with the aid of inequality (1), item 1, § 3, Ch. 7 \((^5)\).

Corollary 2. If \(G\) is Montel, then \(C=B\).

It is enough to prove that \(B\subseteq C\) (Theorem 1). For every \(a\in B\), \(\mathcal D\in \mathbf D\), the family \(a\mathcal D\) is summable in the weakened topology \(\bigl((^5)\), Ch. 4, § 6, Theorem 1, corollary; § 7, item 2, Theorem 3, corollary\()\) and, consequently, summable \(\bigl((^4)\), Ch. 4, § 3, item 4, Proposition 6\()\).

Corollary 3. If \(G\) is a weakly sequentially complete Banach space, then \(C=B\).

It is enough to prove that \(B\subseteq C\). For every \(a\in B\), \(\mathcal D\in \mathbf D\), the family \(a\mathcal D\) is summable in the weakened topology. Every sequence \((aD_n)\), for which \(\{aD_n\}=a\mathcal D\), converges by subsequences in the weakened topology \(\bigl((^3)\), Ch. 3, § 4, item 7, Proposition 9; \((^6)\), Ch. 4, § 1, item 1d\()\) and, consequently, converges by subsequences \(\bigl((^6)\), Ch. 4, § 1, Theorem 1\()\). Consequently, \(a\mathcal D\) is summable \(\bigl((^3)\), Ch. 3, § 4, item 7, Proposition 9; \((^6)\), Ch. 4, § 1, item 1c\()\).

Corollary 4. If \(\mathcal R\) is closed \((S)\), then \(C=B=A\).

This follows directly from the definitions, Theorem 1 and the corollary to Proposition 5, § 2, \((^1)\).

5. Remarks

Remark 1. There exist \(\mathcal R,G\) such that \(V\ne C\).

Let \(U\) be the set of all natural numbers, \(\mathcal R\) the class of all subsets of the set \(U\), and \(G\) an infinite-dimensional Banach space. Consider the mapping \(a'\colon n\to a'n\), where \(|a'n|=1/n\) and \((a'n)_{n\in U}\) is summable \(\bigl((^6)\), Ch. 4, § 1, Theorem 2\()\). For each \(X\in\mathcal R\) define
\[ aX=\sum (a'n)_{n\in X}\quad \bigl((^3),\ \text{Ch. 3, § 4, item 3, Proposition 2}\bigr). \]
Then \(a\in C,\ a\notin V\).

Remark 2. There exist \(\mathcal R,G\) such that \(C\ne B\).

The following example was proposed by S. V. Nagaev. Let \(U\) be the set of natural numbers, \(\mathcal R\) the class of all finite subsets of the set \(U\), and \(G\) the Banach space of all continuous functions on the interval \([0,1]\). Consider the mapping \(a'\): \(a'n=\{X\to X^n-X^{n+1}\}\). For each \(X\in\mathcal R\) define
\[ aX=\sum (a'n)_{n\in X}; \]
then \(a\in B,\ a\notin C\).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
28 VI 1965

REFERENCES

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