MATHEMATICS

D. A. VLADIMIROV

ON THE EXISTENCE OF INVARIANT MEASURES ON BOOLEAN ALGEBRAS

(Presented by Academician V. I. Smirnov on 10 III 1964)

In this note we consider the following problem. Let \(X\) be a complete Boolean algebra, and let \(\mathfrak A\) be a group of its automorphisms.* What properties of the algebra \(X\), on the one hand, and of the collection of automorphisms \(\mathfrak A\), on the other, guarantee the existence on \(X\) of a measure invariant with respect to all automorphisms of the group \(\mathfrak A\)? This problem was partly touched upon by us in the note \((^1)\), in which one may also find explanations concerning the terms and notation occurring below.

As in \((^1)\), by a measure we mean a real-valued function \(\mu\) defined on \(X\) and possessing the following properties:

1) \(\mu(x) > 0\) for \(x \ne 0\);

2) \(\mu(x + y) = \mu(x) + \mu(y)\) for \(x \wedge y = 0\);

3) \(\mu(x_n) \downarrow 0\) when \(x_n \downarrow 0\);

4) \(\mu(1) = 1\).

Properties 2) and 3) together mean the countable additivity of the measure. A function \(\mu\) nonnegative for all \(x\) and possessing properties 2) and 4) will be called a quasimeasure. Invariance of a measure or quasimeasure consists in the fact that, for all \(x \in X\), \(A \in \mathfrak A\),
\[ \mu(Ax)=\mu(x). \]

Let us now enumerate those properties of the collection of automorphisms \(\mathfrak A\) which will be important for us.

\(\mathrm{A}_1\). For all \(x \ne 0\),
\[ \sup_{A\in\mathfrak A} Ax = 1. \]
(This is an ergodicity property of \(\mathfrak A\); it means that there are no elements fixed under all \(A\) other than \(0\) and \(1\).)

A stronger requirement is contained in the condition

\(\mathrm{A}_2\). For every \(x \ne 0\) there exists a finite number of automorphisms \(A_1,A_2,\ldots,A_n \in \mathfrak A\) such that
\[ \sup_{k=1,\ldots,n} A_k x = 1. \]

Properties \(\mathrm{A}_1\) and \(\mathrm{A}_2\) say that the group \(\mathfrak A\) is “sufficiently rich” in automorphisms. We now pass to the enumeration of properties having the opposite character. We shall agree to write \(x \simeq y\) if \(y = Ax\) for some \(A \in \mathfrak A\). Further, if there exist representations of \(x\) and \(y\) in the form of sums of disjoint elements
\[ x=\sum_{\alpha} x_{\alpha},\qquad y=\sum_{\alpha} y_{\alpha}, \]
where \(x_{\alpha}\simeq y_{\alpha}\), then we shall write \(x \sim y\). Here the cardinality of the set of summands may be arbitrary. In the case when the sums are finite, we shall write \(x \simeq y\). Finally, we shall agree to denote by an arrow \(\to\) convergence in the order topology (i.e., in the \((o)\)-topology; see \((^2)\)).

\(\mathrm{B}_1\). If \(x_n \to 0\), then \(A_n x_n \to 0\), whatever \(A_n \in \mathfrak A\) may be (“uniform continuity” of all automorphisms of the group).

\(\mathrm{B}_2\). If \(x_n \simeq x_{n+1}\), \(n=1,2,\ldots\), and \(x_n \wedge x_m = 0\), \(n \ne m\), then
\[ x_1=x_2=\cdots=0. \]

* As usual, by an automorphism we mean a one-to-one mapping of \(X\) onto itself that preserves order. The necessary information from the theory of partially ordered sets is contained, for example, in the monograph \((^2)\).

B₃. If \(x_n=\sum_{\alpha} x_{n\alpha}\) and \(x_n\to 0\), then for any \(A_{n\alpha}\in \mathfrak A\) one has
\[ \sup_{\alpha} A_{n\alpha}x_{n\alpha}\to 0, \]
where the index \(\alpha\) ranges over an arbitrary set depending on \(n\).

B₃⁻. If \(x_n\sim x_{n+1}\), \(n=1,2,\ldots\), and \(x_n\wedge x_m=0\), \(n\ne m\), then
\[ x_1\sim x_2=\cdots=0. \]

B₃″. If \(x_n\simeq x_{n+1}\), \(n=1,2,\ldots\), and \(x_n\wedge x_m=0\), \(n\ne m\), then
\[ x_1=x_2=\cdots=0. \]

B₄. The relations \(x<y\) and \(x\sim y\) are incompatible.

B₅. For every \(x\ne 0\) there exists an invariant quasimeasure \(\mu\) such that
\[ \mu(x)>0. \]

B₆. If
\[ \sum_k x_k\le 1, \]
then for any \(A_k\in\mathfrak A\) one has
\[ \sum_k A_kx_k<+\infty \]
in the extended \(K\)-space “built over” the algebra \(X\).

Finally, let us add to the listed conditions one more, of a purely algebraic character.

K. On the group \(\mathfrak A\) there exists a Banach mean. This condition is satisfied, for example, by every commutative or even solvable group (see on this matter \((^3)\); \((^4)\), p. 141).

Theorem 1. The relation among the conditions B₁, …, K is described by the following diagram:

Here a heavy arrow means “implies in every algebra,” while a light arrow means “implies in a regular algebra.”

We shall agree to denote by \(m\) the assertion: “there exists an invariant measure”; and by \(km\) the assertion: “there exists an invariant quasimeasure possessing property 1).”

Theorem 2. The relation of the conditions A₁, …, B₆ to the assertions \(m\), \(km\) is described by the following diagrams:

Here the heavy and light arrows have the former meaning; a dotted arrow means “implies in a normalized algebra” (i.e. in an algebra on which there exists some measure with properties 1)–4)); \(\&\) is the logical sign of conjunction.

Let us note the following special cases.

a) The fact that conditions A₁ and B₃ together imply the existence of an invariant measure constituted the content of Theorem 1 of the paper \((^1)\). Algebras with a group of automorphisms having properties A₁ and B₃ were called completely homogeneous in \((^1)\). There the general form of such an algebra was also established. Theorem 2 shows,

that in this definition \(B_3\) may be replaced by \(B_4\) (although the equivalence of \(B_3\) and \(B_4\) was established by us in Theorem 1 only for a regular algebra).

b) Theorem 2 shows that in a normed algebra the existence of an invariant measure is ensured by conditions \(K\) and \(B_2\). In particular, when the group consists of powers of a single automorphism, we obtain, as a consequence, the Hajian–Kakutani theorem on an invariant measure (see \((^5)\)).

Leningrad State University
named after A. A. Zhdanov

Received
26 II 1964

REFERENCES

1. D. A. Vladimirov, DAN, 146, 987 (1962).
2. B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Moscow, 1961.
3. G. M. Adelson-Velskii, Yu. A. Shreider, UMN, 12, no. 6 (1957).
4. N. Bourbaki, Topological Vector Spaces, Moscow, 1959.
5. P. Halmos, Sborn. per. Matematika, 6, 3 (1962).