V. G. VINOKUROV

ON INFINITE PRODUCTS OF LEBESGUE SPACES

(Presented by Academician A. N. Kolmogorov on 16 IV 1964)

Lebesgue spaces considered in this paper will be assumed to have no atoms, i.e., to be isomorphic to the unit interval with the usual Lebesgue measure on it.

Let a family \(\{E_t\}\) of Lebesgue spaces be given, where \(t\) ranges over a set \(T\) of cardinality \(\tau\). Denote by \(E^\tau\) the space whose points are all possible systems \((x_t)\), \(x_t \in E_t\). For a measurable set \(A \in E_{t'}\), let \(\bar A=\{(x_t): x_{t'}\in A\}\), and let \(S_{t'}\) be the \(\sigma\)-algebra of sets \(\bar A\) for \(A\in E_{t'}\). Suppose that, for any \(t_1,t_2,\ldots,t_n\), on the smallest \(\sigma\)-algebra containing the \(\sigma\)-algebras \(S_{t_1}, S_{t_2},\ldots,S_{t_n}\), a measure is given such that for \(A\in S_{t_i}\), \(m\bar A=mA\). If all these measures are consistent (see (1)), then they extend to a measure \(m\), defined on the smallest \(\sigma\)-algebra \(Q\) containing all \(\sigma\)-algebras \(S_t\), \(t\in T\). Completing this measure in the usual way, we obtain a measure space, which we shall call the product of the spaces \(E_t\). If for any \(t_1,t_2,\ldots,t_n\) and measurable sets \(A_1\subset E_{t_1}, A_2\subset E_{t_2},\ldots,A_n\subset E_{t_n}\)

\[ m\left(\bigcap_{i=1}^{n}\bar A_i\right)=\prod_{i=1}^{n}mA_i, \]

then the product of the spaces \(E_t\) will be called the direct product and will be denoted by \(\Omega^\tau\).

The metric weight of a measure space \(E\) is the least cardinal number \(\tau\) for which there exists a set \(\mathfrak B\) of cardinality \(\tau\) of measurable subsets of \(E\) such that, for every measurable set \(A\subset E\), there exists a countable sequence \(A_1,A_2,\ldots,A_n,\ldots\) of sets from \(\mathfrak B\) for which

\[ m((A\cap \bar A_n)\cup(\bar A\cap A_n))\to 0. \]

A space \(E\) is called homogeneous if each of its measurable subsets of positive measure, considered as a measure space, has metric weight equal to the metric weight of the space \(E\).

D. Maharam \((^2)\) showed that the metric structure of a homogeneous space of metric weight \(\tau\) is isomorphic to the metric structure of the space \(\Omega^\tau\). In connection with this there arises the problem of describing the class of measure spaces for which not only their metric structures but the spaces themselves are isomorphic to the space \(\Omega^\tau\). A partial solution of this problem is given by the following theorem.

Theorem 1. Every product \(E^\tau\) of Lebesgue spaces which is a homogeneous space with metric weight \(\tau\) is isomorphic to the direct product \(\Omega^\tau\).

Let, for the \(\sigma\)-algebra \(S\), \(\zeta(S)\) be a partition, each element of which either is contained in, or has empty intersection with, any set from \(S\), and for any two elements of \(\zeta(S)\) there is a set from \(S\) containing one element and not containing the other.

A $\sigma$-algebra $S$ is called Lebesgue if the quotient space with respect to the partition $\xi(S)$ is a Lebesgue space. For a certain collection of $\sigma$-algebras $S_\gamma$ we denote by $[S_\gamma]$ the $\sigma$-algebra consisting of all sets from $Q$ that are sums of elements of the partition $\prod_\gamma \xi(S_\gamma)$. Two $\sigma$-algebras $S_1$ and $S_2$ are called independent if, for any $A \in S_1$ and $B \in S_2$,
$m(A \cap B)=mA\cdot mB$.

In the proof of Theorem 1 the following two lemmas are used.

Lemma 1. Let a set $T' \subset T$ have cardinality less than $\tau$, and let $R \subset T\setminus T'$ be a countable set.

There exists a Lebesgue $\sigma$-algebra $S \subset Q$, independent of $[S_t,\, t \in T']$, and such that
\[ [S_t,\, t \in R]\subset [S,\, S_t,\, t \in T']. \]

The proof of this lemma is a slight modification of the proof of Theorem No. 3, § 4, from the paper (³).

Lemma 2. Let $\xi_1$ and $\xi_2$ be two measurable partitions of a Lebesgue space whose product is the partition into points, and suppose that every element of the partition $\xi_1$ is a set of cardinality continuum.

Then there exists a measurable partition $\xi'_2=\xi_2\pmod 0$ such that the product $\xi_1\times \xi'_2$ is the partition into points and every element of $\xi_1$ has nonempty intersection with every element of $\xi'_2$.

To prove Theorem 1, arrange the set $T$ into a transfinite sequence
\[ t_1,t_2,\ldots,t_r,\ldots, \]
where $r$ runs through the set of transfinite numbers less than $\omega_\tau$—the first transfinite number of cardinality $\tau$, and put $S'=S_{t_1}$.

Suppose that, for some $\alpha<\omega_\tau$, Lebesgue $\sigma$-algebras $S^r\subset Q$, $1\le r\le \alpha-1$, have been constructed with the following properties:

1) For any two groups of indices $r_1,\ldots,r_n$ and $r',\ldots,r^k$, the $\sigma$-algebras
\[ [S^{r_1},\ldots,S^{r_n}] \quad\text{and}\quad [S^{r'},\ldots,S^{r^k}] \]
are independent.

2) There exists a set $T_{\alpha-1}\subset T$, whose cardinality is equal to the cardinality of $\alpha-1$, containing all $t_r$, $r\le \alpha-1$, and such that
\[ Q(\alpha-1)=[S^r,\, r\le \alpha-1]=[S^t,\, t\in T_{\alpha-1}]. \]

3) Every intersection
\[ \bigcap_{1\le r\le \alpha-1} C_r, \]
where $C_r$ is an element of the partition $\xi(S^r)$, is nonempty.

Using Lemma 1, construct countable sequences of Lebesgue $\sigma$-algebras
\[ S_1,S_2,\ldots,S_n,\ldots,\qquad S_n\in Q, \]
and countable sets
\[ R_0=(t_\alpha),\quad R_1\subset T\setminus (T_{\alpha-1}\cup R_0),\ldots,\quad R_n\subset T\setminus\left(T_{\alpha-1}\cup\left(\bigcup_{i=0}^n R_i\right)\right),\ldots \]
such that
\[ [Q(\alpha-1),S_1]\supset S_{t_\alpha},\qquad [Q(\alpha-1),S_1,\ldots,S_n]\supset [S_t,\, t\in R_{n-1}], \]
\[ S_n\subset\left[Q(\alpha-1),S_t,\, t\in\bigcup_{i=1}^n R_i\right] \]
and $S_n$ is independent of $[Q(\alpha-1),S_1,\ldots,S_{n-1}]$.

(If $t_\alpha\in T_{\alpha-1}$, then $R_0$ is empty, and $S_1$ is any Lebesgue $\sigma$-algebra independent of $Q(\alpha-1)$.) Let
\[ S_\alpha=[S_1,S_2,\ldots,S_n,\ldots], \]
\[ T_\alpha=T_{\alpha-1}\cup\left(\bigcup_{n=0}^\infty R_n\right), \]
and let $R\subset T_{\alpha-1}$ be such a countable set that
\[ [S_t,\, t\in R\cup(\bigcup_{n=0}^\infty R_n)]\supset S_\alpha. \]
Let $S=[S_t,\, t\in R]$.

The quotient space with respect to the partition $\xi(S)\times\xi(S_\alpha)$ is a Lebesgue space, and the partitions $\xi(S)$ and $\xi(S_\alpha)$ induce in it partitions $\xi_1$ and $\xi_2$,

satisfying the conditions of Lemma 2. Let \(\xi\) be the partition of the space \(E^\tau\) induced by the partition \(\xi'_2\), whose existence is asserted by this lemma. Denote by \(S^\alpha\) some \(\sigma\)-algebra of sets from \(Q\) for which \(\xi(S^\alpha)=\xi\). The \(S^\alpha\), together with the previously constructed \(S^r\), satisfy conditions 1), 2), and 3). If, however, \(\alpha\) is a limit ordinal and the \(S^r\) have been constructed for all \(r<\alpha\), then \(S^\alpha\) is constructed in the same way as above, but instead of \(Q(\alpha-1)\) and \(T_{\alpha-1}\) one takes \(Q(\alpha-)=[S^r,\ r<\alpha]\) and \(T_{\alpha-}=\bigcup_{r<\alpha}T^r\). After the completion of this process the isomorphism between \(E^\tau\) and \(\Omega^\tau\) is obvious.

Let \(E\) be a topological space on which a nonatomic measure \(m\) is given, satisfying the following conditions:

1. \(mE=1\).
2. The measure space \(E\) is homogeneous.
3. There exists a system \(\Sigma\) of measurable sets, whose cardinality is equal to the metric weight of the space \(E\), such that every measurable set \(A\) has the form \(A=B\cup A_0,\ A_0\subset B_0\), where \(B\) and \(B_0\) belong to the smallest \(\sigma\)-algebra containing all the sets of the system \(\Sigma\), \(mB_0=0\), and for any two points of \(E\) there is a set in \(\Sigma\) containing one point and not containing the other.

All these conditions are necessary in order that the space \(E\) be isomorphic to \(\Omega^\tau\). The following question seems interesting: how must the measure \(m\) be related to the topology of \(E\) in order that the space \(E\) be isomorphic to \(\Omega^\tau\)? Let us write down three more conditions.

1. For every measurable set \(A\),
   \[ mA=\sup_{F\subset A} mF, \]
   where \(F\) ranges over measurable bicompact sets contained in \(A\).

Let \(Q\) be the smallest \(\sigma\)-algebra containing all measurable bicompact sets.

1. \(Q\) contains a system \(\Sigma\) possessing property 3.
2. For any Lebesgue \(\sigma\)-algebra \(Q'\subset Q\) there is a Lebesgue \(\sigma\)-algebra \(Q''\supset Q'\), \(Q''\subset Q\), such that all elements of the partition \(\xi(Q'')\) are bicompact sets and each element of the partition \(\xi(Q')\) contains at least two elements of the partition \(\xi(Q'')\).

With the help of Theorem 1 one can prove the following theorem.

Theorem 2. If a topological space with measure has metric weight continuum and satisfies conditions 1–6, then it is isomorphic to the space \(\Omega^\tau\), where \(\tau\) is the continuum.

Institute of Mathematics named after V. I. Romanovskii
Academy of Sciences of the Uzbek SSR

Received
30 III 1964

REFERENCES

1. A. N. Kolmogorov, Basic Concepts of Probability Theory, Moscow–Leningrad, 1936.
   D. Maharam, Proc. Nat. Acad. Sci., 28, 108 (1942).
   V. A. Rokhlin, Matem. sborn., 25, no. 1, 107 (1949).