UDC 513.881

MATHEMATICS

I. A. BEREZANSKII

ON LIMITS OF MOLECULAR MEASURES IN SOME SPACES

(Presented by Academician P. S. Novikov on 4 IV 1966)

In the works of Katětov \((^{1-3})\), Raikov \((^4)\), and Ptak \((^5)\), the following general scheme was studied. On a set \(X\) one considers a certain linear space \(P(X)\) of scalar functions separating the points of \(X\) (the field of scalars is \(R\) or \(C\)). Let \(L(X)\) be the free linear space of the set \(X\) over the given field of scalars. The elements of \(L(X)\) are formal linear combinations \(u=\sum_{x\in X}\lambda_x x\), where the \(\lambda_x\) are scalar coefficients different from zero for only a finite number of points of \(X\). The expression \(\langle u,f\rangle=\sum_{x\in X}\lambda_x f(x)\) defines on the product \(L(X)\times P(X)\) a bilinear form, by means of which a duality is established between \(L(X)\) and \(P(X)\). Using this duality, one can topologize \(L(X)\) in various ways.

But the elements of the space \(L(X)\) may be treated as measures on the set \(X\), concentrated at a finite number of points \(x_1,\ldots,x_n\) and equal there, respectively, to \(\lambda_{x_1},\ldots,\lambda_{x_n}\). Such measures we shall call molecular. In the present note we investigate the possibility of approximating a sufficiently broad class of measures by means of molecular measures under a suitably chosen topology, as well as the question of how far all objects obtained in this approximation process may be identified with measures.

We shall say that a system \(\mathcal H=\{H_\alpha\}_{\alpha\in A}\) of subsets of \(P(X)\) is \(b\)-closed if: 1) it forms a covering of the space \(P(X)\); 2) it is closed with respect to the operations of taking subsets, homotheties of a set, the algebraic sum of two sets, and taking the absolutely convex hull of a set; 3) for every \(x\in X\) and every \(\alpha\in A\) the set \(\{f(x)\}_{f\in H_\alpha}\) is bounded. If the system \(\mathcal H\) is \(b\)-closed, then the system of sets
\[ U_{\alpha,\varepsilon}=\{u\in L(X): |\langle u,f\rangle|<\varepsilon \text{ for all } f\in H_\alpha\} \]
\((\alpha\in A,\ \varepsilon>0)\) forms a base of neighborhoods of zero in a separated locally convex topology \(\tau_{\mathcal H}\) on \(L(X)\).

Let on the set \(X\) there be singled out a certain class \(M\) of scalar measures, containing all molecular measures, forming a linear space and such that for every measure \(\mu\in M\) and every function \(f\in P(X)\) there exists the integral \(\int_X f\,d\mu\), and moreover the set \(\left\{\int_X f\,d\mu\right\}_{f\in H_\alpha}\) is bounded for every \(\alpha\in A\). In that case the system of sets
\[ \widetilde U_{\alpha,\varepsilon} = \{\mu\in M: |\int_X f\,d\mu|<\varepsilon \text{ for all } f\in H_\alpha\} \]
\[ (\alpha\in A,\ \varepsilon>0) \]
forms a base of neighborhoods of zero in a separated locally convex topology \(\tau_{\mathcal H}\) on \(M\), and \((L(X),\tau_{\mathcal H})\) is a topological linear subspace of \((M,\tau_{\mathcal H})\). If, moreover, the completion \(\widehat L\) (or the bounded completion \(\widehat L\)) of the space \((L(X),\tau_{\mathcal H})\) coincides

as a linear space with \(M\), then we shall say that \(\hat L\) (respectively, \(\check L\)) is canonically isomorphic to \(M\). Under this isomorphism, to an element \(\hat u \in \hat L\) (respectively, \(\check L\)) there corresponds a measure \(\mu \in M\) such that
\[ \langle \hat u, f\rangle=\int_X f\,d\mu \]
for every function \(f \in P(X)\).

As usual, for a measure \(\mu\) defined on some \(\sigma\)-algebra of subsets of \(X\), by \(|\mu|(B)\) we shall denote the variation of \(\mu\) on the set \(B\), and we shall call the measure bounded if \(|\mu|(X)<\infty\). A Borel measure \(\mu\) on a completely regular topological space \(X\) will be called regular if for every Borel set \(B\) (in particular, for all of \(X\))
\[ |\mu|(B)=\sup_{T\subset B}|\mu|(T), \]
where \(T\) are bicompact sets. The notation \(Rb(X)\) is used for the space of all bounded regular Borel measures on \(X\).

Theorem 1. Let \(X\) be bicompact. Let \(P(X)\) be taken to be the space \(C(X)\) of all continuous functions on \(X\), and let \(\mathcal H\) be any \(b\)-closed family of \(\sigma(c,c')\)-bicompact sets containing all sequences uniformly converging to zero. In that case the completion \(\hat L\) of the space \((L(X),\tau_{\mathcal H})\) is canonically isomorphic to \(Rb(X)\).

Remark. Special cases of this theorem were considered by Katetov \((^2,^3)\).

We shall say that a set \(H\) of continuous functions on a topological space \(X\) has the \(k\)-property if it is uniformly bounded on \(X\) and quasi-equicontinuous \((^6)\) on every bicompact \(T\subset X\). If the original space \(X\) is bicompact, this is equivalent to \(\sigma(c,c')\)-bicompactness of the set \(H\) \((^6)\). If \(X\) is a \(k\)-space, then this induces bicompactness of the set \(H\) in the topology of pointwise convergence on \(X\).

Theorem 2. Let \(X\) be a completely regular space. Let \(P(X)\) be taken to be any subspace of the space of all bounded continuous functions on \(X\) such that, for every bicompact \(T\subset X\), the restrictions of all functions \(f\in P(X)\) to \(T\) form the entire space \(C(T)\). Let \(\mathcal H\) be a \(b\)-closed system of subsets of \(P(X)\) having the \(k\)-property. Then \(Rb(X)\subset \hat L\), where \(\hat L\) is the completion of the space \((L(X),\tau_{\mathcal H})\).

Consequence. Let \((X,\xi)\) be a separated uniform space, \(P(X)\) the space of all bounded uniformly continuous functions on \(X\). Let \(\mathcal H\) be a \(b\)-closed system of subsets of \(P(X)\) having the \(k\)-property. Then \(Rb(X)\subset \hat L\).

We shall say that a measure \(\mu\in Rb(X)\) is \(\tau_{\mathcal H}\)-approximable if there exists a sequence \(u_1,\ldots,u_n,\ldots\) of elements of \(L(X)\), \(\tau_{\mathcal H}\)-converging to \(\mu\).

Theorem 3. Let \(X\) be a completely regular space such that every bicompact \(T\subset X\) is metrizable. Let \(P(X)\) satisfy the conditions of Theorem 2; let \(\mathcal H\) be a \(b\)-closed system, each set in which is uniformly bounded on \(X\) and equicontinuous at every point \(x\in X\). Then every measure \(\mu\in Rb(X)\) is \(\tau_{\mathcal H}\)-approximable.

Theorem 4. Let \(X\) be taken to be the finite-dimensional Euclidean space \(E^n\). Let \(P_i(E^n)\) be the space of all bounded uniformly continuous functions on \(E^n\). Let \(\mathcal H\) be a \(b\)-closed system of subsets of \(P(E^n)\) such that every set \(H\in\mathcal H\) is uniformly bounded and uniformly equicontinuous on \(E^n\), and moreover all countable sets of the indicated kind are contained in \(\mathcal H\). In that case the completion \(\hat L(E^n)\) of the space \((L(E^n),\tau_{\mathcal H})\) is canonically isomorphic to the space \(Rb(E^n)\).

If \((X,\xi)\) is a separated uniform space; \(P(X)\) is the space of all uniformly continuous functions on \((X,\xi)\); \(\mathcal H\) is a family

of all subsets of \(P(X)\), pointwise bounded on \(X\) and equicontinuously uniformly continuous, then \((L(X),\tau_{\mathcal H})\) is a free locally convex space of the original space \((X,\xi)\) \((^4)\).

A measure \(\mu \in Rb(E^n)\) is said to have a first moment if
\[ \int_X |x|\,d\mu < \infty, \]
where \(|x|\) is the Euclidean norm of the element \(x \in E^n\). Denote by \(\widetilde{Rb}(X)\) the space of all such measures.

Theorem 5. The completion of the free locally convex space of a finite-dimensional Euclidean space \(E^n\) is canonically isomorphic to the space \(\widetilde{Rb}(X)\).

Recall that the bounded completion of a topological linear space \((E,\tau)\) is the smallest boundedly complete (in other terminology, quasi-complete \((^7)\)) space \((\breve E,\breve\tau)\) containing \((E,\tau)\) as an everywhere dense topological and linear subspace.

A uniform space \((X,\xi)\) is called uniformly locally bicompact if its uniformity \(\xi\) has a base \(v\) of entourages such that, for every \(V \in v\) and for every \(x \in X\), the set \(V[x]\) is bicompact.

Theorem 6. Let \((X,\xi)\) be a separable uniformly locally bicompact space; \(P(X)\) the space of all bounded uniformly continuous functions on \((X,\xi)\); \(\mathcal H\) the family of all uniformly bounded equicontinuously uniformly continuous sets from \(P(X)\). In this case the bounded completion \(\breve L\) of the space \((L(X),\tau_{\mathcal H})\) is canonically isomorphic to the space \(Rb(X)\).

Theorem \(6'\). Let \(X\) be a paracompact locally bicompact topological space; \(P(X)\) the space of all bounded continuous functions on \(X\); \(\mathcal H\) the family of all subsets of \(P(X)\) uniformly bounded on \(X\) and equicontinuous at every point \(x \in X\). In this case the bounded completion \(\breve L\) of the space \((L(X),\tau_{\mathcal H})\) is canonically isomorphic to the space \(Rb(X)\).

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
1 IV 1966

CITED LITERATURE

\(^1\) M. Katětov, Proc. Intern. Congr. Math., 1962, p. 473.
\(^2\) M. Katětov, XX Simp. di topologia, 1964.
\(^3\) M. Katětov, Comment. Math. Univ. Carolin., 6, 1965, p. 251.
\(^4\) D. A. Raikov, Matem. sborn., 63, No. 4, 582 (1964).
\(^5\) V. Pták, Czechoslovak Math. J., 14 (89), 562 (1964).
\(^6\) N. Danford, J. Schwartz, Linear Operators, Moscow, 1963.
\(^7\) N. Bourbaki, Topological Vector Spaces, III, 1958.