MATHEMATICS

V. G. VINOKUROV

ON THE CONTINUAL PRODUCT OF LEBESGUE SPACES

(Presented by Academician A. N. Kolmogorov on 1 XII 1964)

Denote by \(\tau\) the cardinality of the continuum, and by \(E^\tau\) the direct product of a continuum of unit intervals with the usual Lebesgue measure on them. In this note we consider conditions under which a space with measure is isomorphic to \(E^\tau\).

Let \(E\) be a space with measure \(m\), defined on a \(\sigma\)-algebra of measurable sets, such that the following conditions are satisfied:

1) \(mE = 1;\)
2) if \(A \subset B\), where \(B \in Q\) and \(mB = 0\), then \(A \in Q\).

The metric weight of the space \(E\) is the least of the cardinal numbers \(\lambda\) for which there exists a system \(\mathfrak{B}\) of cardinality \(\lambda\) of measurable sets such that, for every measurable set \(A\), there is a countable sequence \(A_1, A_2, \ldots, A_n, \ldots\) of sets from \(\mathfrak{B}\) for which

\[ m\bigl((A \cap \overline{A}_n)\cup(\overline{A}\cap A_n)\bigr)\to 0. \]

The space \(E\) is called homogeneous if every measurable subset of it of positive measure, considered as a space with measure, has metric weight equal to the metric weight of the space \(E\).

The exact weight of the space \(E\) is the least of the cardinal numbers \(\lambda\) for which there exists a system \(\Sigma\) of cardinality \(\lambda\) of measurable sets such that:

a) every measurable set \(A\) has the form

\[ A = A^1 \cup A^2, \]

where \(A^1\) belongs to the smallest \(\sigma\)-algebra \(H\) containing \(\Sigma\), and \(A^2 \subset B\), where \(B \in H\) and \(mB = 0\);

b) for any two points of \(E\) there is a set in \(\Sigma\) containing one point and not containing the other. A system \(\Sigma\) having properties a) and b) and having cardinality equal to the exact weight of the space will be called a basis of \(E\).

A space isomorphic to \(E^\tau\) must satisfy, in addition to 1) and 2), the following conditions:

3) the space \(E\) is homogeneous;
4) the metric weight of \(E\) is equal to the exact weight of \(E\) and is equal to \(\tau\).

Finally, the last condition is as follows:

5) there exists a class \(\Phi = \{F\}\) of subsets of the space \(E\) satisfying the requirements:

a) every centered countable sequence of sets from \(\Phi\) has a nonempty intersection;

b) the intersection of a countable number of sets from \(\Phi\), if it is nonempty, belongs to \(\Phi\);

c) each set from \(\Phi\) has cardinality \(2^\tau\);

d) for every measurable set \(A\) of positive measure, set

\[ mA=\sup_{\substack{F\in\Phi,\; F\subset A}} m_xF, \]

where

\[ m_xF=\sup_{\substack{F\in Q,\; B\subset F}} mB. \]

There exists an example of a space with measure that is not isomorphic to the space \(E^t\), but satisfies conditions 1)—4) and the compactness condition of the measure in the sense of Marczewski \((^1)\), which consists in the fact that in \(E\) there exists a class \(\Phi\) satisfying requirements a), b), and d) of condition 5.

This example can also be made so that the class \(\Phi\) will satisfy requirements a), c), and d), but will not satisfy requirement b).

Theorem. A space with measure satisfying conditions 1)—5) is isomorphic to the space \(E^t\).

Proof. For a system \(R\) of sets, denote by \(\zeta(R)\) the partition of the space \(E\), every element of which either is contained in, or has nonempty intersection with, every set from \(R\), and for any two elements of \(\zeta(R)\) there is a set from \(R\) containing one element and not containing the other.

We shall call a \(\sigma\)-algebra \(R\subset Q\) Lebesgue if the quotient space with respect to the partition \(\zeta(R)\) is isomorphic to the unit interval with Lebesgue measure. Let \(R\) be a Lebesgue \(\sigma\)-algebra, \(\Sigma'_R\) a basis of the quotient space with respect to the partition \(\zeta(R)\), and \(\Sigma_R\) the system of sets that are inverse images of sets from \(\Sigma'_R\) under the mapping \(T_{\zeta(R)}\) (\(T_\zeta\) is the mapping of \(E\) onto \(E/\zeta\), sending each point of \(E\) to that element of the partition \(\zeta\) in which it is contained).

Construct a countable algebra \(\mathfrak B\) of measurable sets such that:

a) \(\Sigma_R\subset \mathfrak B\);

b) for every set \(A\in\mathfrak B\) of positive measure and every \(\varepsilon>0\), there is a \(B\in\mathfrak B\) such that \(B\subset A\), \(m(A\setminus B)<\varepsilon\), and there exists a set \(F\in\Phi\) for which \(B\subset F\subset A\).

Denote by \(S(R)\) the smallest \(\sigma\)-algebra containing \(\mathfrak B\). It is easy to see that almost all elements of the partition \(\zeta(S(R))\) belong to \(\Phi\).

Take some basis \(\Sigma\) of the space \(E\) and arrange the sets from \(\Sigma\) into a transfinite sequence

\[ Z_1,\ldots,Z_\lambda,\ldots,\quad \lambda<\omega_\tau . \]

Now take some Lebesgue \(\sigma\)-algebra \(R'\) containing \(Z_1\), and construct \(S_1=S(R)\). Put \(S'=S_1\), and for all \(\lambda<\omega_\tau\) construct Lebesgue \(\sigma\)-algebras \(S^\lambda\) such that:

a) if \(\lambda'<\lambda''\), then \(S^{\lambda'}\subset S^{\lambda''}\);

b) \(Z_\lambda\in S^\lambda\);

c) almost all elements of the partition \(\zeta(S^\lambda)\) belong to \(\Phi\);

d) every element of the partition

\[ \prod_{\gamma<\lambda}\zeta(S^\gamma), \]

belonging to \(\Phi\), contains a continuum of elements of the partition \(\zeta(S^\lambda)\);

e) there exists a null set \(M_\lambda\in S^\lambda\) which intersects each element of the partition

\[ \prod_{\gamma<\lambda}\zeta(S^\gamma), \]

having cardinality \(2^\tau\), in a set of cardinality \(2^\tau\).

We shall now use the following proposition, which is a slight generalization of one lemma from \((^2)\). Let \(\zeta_1\) and \(\zeta_2\) be two measurable partitions of a Lebesgue space, whose product is the partition into points. Then there exists a measurable partition \(\zeta'_2=\zeta_2\pmod 0\) such that the product \(\zeta_1\times\zeta'_2\) is the partition into points, and every element of \(\zeta_1\) having a continuum of points has a nonempty intersection with every element of \(\zeta'_2\).

With the aid of this proposition, for each \(\lambda\) we construct a Lebesgue \(\sigma\)-algebra \(S_\lambda \subset S^\lambda\) such that every element of the partition belonging to \(\Phi\)

\[ \prod_{\gamma<\lambda}\zeta(S^\gamma) \]

has nonempty intersection with each element of \(\zeta(S_\lambda)\).

Let us now take the space \(\widetilde E\), whose points are all possible sequences of elements of the partitions \(\zeta(S_\lambda)\),

\[ (c^\lambda)=c^1,\ldots,c^\lambda,\ldots,\quad \lambda<\omega_\tau, \]

one taken from each partition. Having defined, for arbitrary sets,

\[ \widetilde A_1=\{(c^\lambda):c^{\lambda_1}\subset A^1\in S_{\lambda_1}\},\ldots,\quad \widetilde A_n=\{(c^\lambda):c^{\lambda_n}\subset A^n\in S_{\lambda_n}\}, \]

\[ m\left(\bigcap_{i=1}^{n}\widetilde A_i\right) = m\left(\bigcap_{i=1}^{n} A^i\right) \]

and continuing in the usual way, we define a measure in \(\widetilde E\). In \((2)\) it is shown that the measure space so defined is isomorphic to the space \(E^\tau\). Therefore it remains only to prove that the space \(E\) is isomorphic to the space \(\widetilde E\). We shall construct an isomorphic mapping of \(E\) onto \(\widetilde E\) as follows. Let \(M^\lambda\) be the sum of all elements of the partition \(\zeta(S^\lambda)\) not belonging to \(\Phi\). Let \(c_0\) be some element of the partition \(\zeta(S_1)\) having cardinality \(2^\tau\). We define

\[ M_{(1)}=M^1\cup c_1\cup c_0,\quad M_{(\lambda)}=M^\lambda\cup M_\lambda\cup c_\lambda,\quad N_\lambda=M_{(\lambda)}\setminus\left(\bigcup_{\gamma<\lambda}M_{(\gamma)}\right). \]

Now define

\[ \widetilde N_\lambda=\{(c^\alpha):\bigcap_{\alpha<\lambda}c^\alpha\in N_\lambda\} \]

and denote by \(U_\lambda\) a one-to-one mapping of \(N_\lambda\) onto \(\widetilde N_\lambda\), under which, for any element \(c\) of the partition

\[ \prod_{\gamma<\lambda}\zeta(S^\gamma), \]

having nonempty intersection with \(N_\lambda\),

\[ U_\lambda(C\cap N_\lambda)=\{(c^\alpha):\bigcap_{\alpha<\lambda}c^\alpha=c\}\cap\widetilde N_\lambda. \]

The mapping of the space \(E\) onto \(\widetilde E\) which coincides on each \(N_\lambda\) with \(U_\lambda\) is an isomorphic mapping of \(E\) onto \(\widetilde E\).

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

Received
13 VI 1964

CITED LITERATURE

1. E. Marczewski, Fund. Math., 40, 113 (1953).
2. V. G. Vinokurov, DAN, 158, No. 6 (1964).