MATHEMATICS

L. Ya. SAVEL’EV

ON CONTINUOUS MEASURES

(Presented by Academician S. L. Sobolev on 22 VI 1964)

General properties are described of the sequential order topology in the Boolean ring \(\mathcal P\) of all subsets of a given set. Measures are considered that are continuous mappings of a subring \(\mathcal A\) of \(\mathcal P\) into a regular Abelian semigroup; conditions are given for the extendability of such measures to the closure of \(\mathcal A\).

1. Consider the set \(\mathcal P=\mathcal P(E)\) of all subsets of the set \(E\). Define in \(\mathcal P\) the ring structure in the usual way: for arbitrary sets \(X\) and \(Y\) in \(\mathcal P\), the sum \(X+Y\) is the symmetric difference of these sets, and the product \(XY\) is their intersection. We order the set \(\mathcal P\) by inclusion. Thus \(\mathcal P\) is a Boolean ring with identity and a complete Boolean algebra \(\left((^1),\right.\) Ch. 2, § 1; Ch. 10, §§ 1–3). Define in \(\mathcal P\) a topology \(\left((^2),\right.\) Ch. 1, § 7) as the strongest of the topologies in \(\mathcal P\) for which, for every sequence \((X_n)\), \(X_n\in\mathcal P\), and every \(A\in\mathcal P\), convergence of \((X_n)\) to \(A\) in order \(\left(X_n \xrightarrow{o} A;\ (^1),\right.\) Ch. 4, § 8) implies convergence of \((X_n)\) to \(A\) \(\left(X_n\to A;\ (^2),\right.\) Ch. 1, § 6, item 4). We shall call the topology thus obtained the \(so\)-topology (cf. \((^3)\)). The set \(\mathcal P\), with the ring, order, and topology structures defined on it in the indicated way, will be denoted by \(\mathcal P_{so}=\mathcal P_{so}(E)\).

The following propositions describe the general properties of the topological space \(\mathcal P_{so}\).

A set \(\mathcal F\subseteq\mathcal P_{so}\) is closed if and only if, for every sequence \((X_n)\), \(X_n\in\mathcal F\), and every \(A\in\mathcal P_{so}\), from \(X_n\xrightarrow{o}A\) it follows that \(A\in\mathcal F\) (cf. \((^4)\), Ch. 1, § 2, item 4).

Theorem 1. For every \(A\in\mathcal P_{so}\), the translation \(Y=A+X\) is a homeomorphism.

Theorem 2. The topological space \(\mathcal P_{so}\) is separated (Hausdorff) and totally disconnected.

Theorem 3. If \(E\) is uncountable, the topological space \(\mathcal P_{so}(E)\) is noncompact.

Theorem 4. The operations \(X+Y\) and \(XY\) in \(\mathcal P_{so}\) are continuous in each variable.

Theorem 5. The closure of a ring \(\mathcal A\) in \(\mathcal P_{so}\) coincides with the \(\sigma\)-ring generated by \(\mathcal A\).

1. Consider a topological Abelian semigroup \(B\) with regular topology and zero, and a subring \(\mathcal A\) of the ring \(\mathcal P_{so}\) with the topology induced from \(\mathcal P_{so}\). A mapping \(m\) of the ring \(\mathcal A\) into the semigroup \(B\) will be called a measure (on \(\mathcal A\) with values in \(B\)) if:

1) \(m(0)=0\),

2) \(m(X+Y)=m(X)+m(Y)\) for all \(X\) and \(Y\) in \(\mathcal A\) such that \(XY=0\).

Condition 2) in the definition of a measure is equivalent to the condition

2′) \(m(X+Y+XY)+m(XY)=m(X)+m(Y)\) for all \(X\) and \(Y\) in \(\mathcal A\).

Measures that are continuous mappings of the space \(\mathcal A\) into the space \(B\) will be called continuous measures. Consider, for example, continuous measures with values in the semigroup \(B=R_{+}\) of all positive (\(\ge 0\)) numbers.

Theorem 6. A positive numerical measure is continuous if and only if it is countably additive and bounded.

Indeed, the continuity of a measure implies its countable additivity. The unboundedness of a measure entails its discontinuity at zero, since in this case there exists a sequence of sets having measures \(\geq 1\) and converging to \(0\). If a measure is countably additive and bounded on a \(\sigma\)-ring, then the preimage of a closed numerical set is closed in \(\mathfrak{P}_{s o}\). Hence the sufficiency of the condition of the theorem follows, since a countably additive bounded measure extends uniquely from the ring to the \(\sigma\)-ring generated by it \(\left({}^{5}\right)\), Ch. 3, § 13, Theorem 1.

Let \(\overline{\mathcal A}\) be the closure of the ring \(\mathcal A\) in \(\mathfrak{P}_{s o}\), and let \(m\) be a continuous measure on \(\mathcal A\) with values in the semigroup \(B\). We give conditions for the extendability of the measure \(m\) from \(\mathcal A\) to \(\overline{\mathcal A}\).

Theorem 7. For the existence of a continuous measure \(\overline m\) on \(\overline{\mathcal A}\) that is an extension of the measure \(m\), it is necessary and sufficient that the limit
\[ \lim_{\substack{X \ni \overline X,\; X \in A}} m(X) \]
exist for all \(\overline X \in \overline{\mathcal A}\).

The condition of the theorem is necessary and sufficient for
\[ \overline m(\overline X) = \lim_{\substack{X \ni \overline X,\; X \in A}} m(X) \]
to be a continuous mapping of \(\overline{\mathcal A}\) into \(B\) \(\left({}^{2}\right)\), Ch. 1, § 6, item 7, Theorem 1). It remains to verify condition \(2'\)) for \(\overline m\). The validity of this condition follows from the continuity of addition in \(B\), the partial continuity of the operations in \(\mathfrak{P}_{s o}\), the continuity of \(\overline m\) on \(\overline{\mathcal A}\), and the fulfillment of the corresponding equality for \(\overline m\) on \(\mathcal A\).

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

Received
17 VI 1964

References

\({}^{1}\) G. Birkhoff, Lattice Theory, 1952.
\({}^{2}\) N. Bourbaki, General Topology (Basic Structures), 1958.
\({}^{3}\) R. Bagley, Michigan Math. J., 3, No. 2, 105 (1956).
\({}^{4}\) L. Kantorovich, B. Vulikh, A. Pinsker, Functional Analysis in Partially Ordered Spaces, 1950.
\({}^{5}\) P. Halmos, Measure Theory, 1953.