Corresponding Member of the USSR Academy of Sciences D. MENSHOV

ON UNIVERSAL SEQUENCES OF FUNCTIONS

A sequence of measurable functions

\[ f_m(x) \quad (m = 0,1,2,\ldots), \tag{1} \]

defined almost everywhere on some interval \([a,b]\), will be called universal* on \([a,b]\) if, for every measurable function \(f(x)\) defined almost everywhere on \([a,b]\), one can choose an increasing sequence of natural numbers \(m_k\) \((k = 0,1,2,\ldots)\) such that

\[ \lim_{k\to\infty} f_{m_k}(x) = f(x) \tag{2} \]

almost everywhere on \([a,b]\).

We next introduce the definition of a sequence universal with respect to two functions \(F_1(x)\) and \(F_2(x)\), or of a sequence of type \(A(F_1,F_2)\).**

Let the functions \(F_1(x)\) and \(F_2(x)\) be measurable, defined almost everywhere on the interval \([a,b]\), and satisfy the inequality

\[ F_1(x) \leqslant F_2(x) \tag{3} \]

almost everywhere on this interval. We shall say that the sequence (1) of measurable functions, defined almost everywhere on \([a,b]\), is a sequence of type \(A(F_1,F_2)\) if, for every measurable function \(f(x)\) defined almost everywhere on \([a,b]\) and satisfying the inequality

\[ F_1(x) \leqslant f(x) \leqslant F_2(x) \tag{4} \]

almost everywhere on the same interval, one can determine an increasing sequence of natural numbers \(m_k\) \((k = 0,1,2,\ldots)\) such that equality (2) will hold almost everywhere on \([a,b]\) ((\(^{1}\)), p. 3).

Let us introduce one more definition. Let the functions \(F_1(x)\) and \(F_2(x)\) satisfy the same conditions as in the preceding definition. We shall say that the sequence (1) of measurable functions, defined almost everywhere on \([a,b]\), is a sequence of type \(B(F_1,F_2)\) if, for any four measurable functions \(\varphi_1(x)\), \(\psi_1(x)\), \(\psi_2(x)\), \(\varphi_2(x)\), defined almost everywhere on \([a,b]\) and satisfying the inequality

\[ F_1(x) \leqslant \varphi_1(x) \leqslant \psi_1(x) \leqslant \psi_2(x) \leqslant \varphi_2(x) \leqslant F_2(x) \tag{5} \]

almost everywhere on the same interval, one can determine an increasing sequence of natural numbers \(m_k\) \((k = 0,1,2,\ldots)\) such that

\[ \alpha^0.\quad \lim_{k\to\infty} f_{m_k}(x) = \varphi_1(x), \qquad \overline{\lim}_{k\to\infty} f_{m_k}(x) = \varphi_2(x) \]

almost everywhere on \([a,b]\).

\[ \beta^0.\quad \psi_1(x)\ \text{and}\ \psi_2(x) \]

are respectively the lower and upper limits in measure on \([a,b]\) of the sequence of functions

\[ f_{m_k}(x) \quad (k = 0,1,2,\ldots). \tag{6} \]

(The definition of upper and lower limits in measure is given in (\(^{2}\)), p. 4.)

* The functions of the sequence (1) may be equal to \(+\infty\) or \(-\infty\) on sets of positive measure.

** In what follows, when speaking of universal sequences, we shall not mention the interval on which they are universal.

An infinite series whose terms are measurable functions, finite almost everywhere on \([a,b]\), will be called universal (respectively, a series of type \(A(F_1,F_2)\) or of type \(B(F_1,F_2)\)) if the sequence of its partial sums is universal (respectively, of type \(A(F_1,F_2)\) or of type \(B(F_1,F_2)\)).

In \((^1)\) the following theorem was proved.

If the functions \(F_1(x)\) and \(F_2(x)\) satisfy the preceding conditions, where \([a,b]=[-\pi,\pi]\), then the classes of trigonometric series of type \(A(F_1,F_2)\) and of type \(B(F_1,F_2)\) coincide \((^1\), Theorem 1, p. 5).

It turns out that this theorem remains valid for arbitrary sequences of measurable functions. One can even prove a more general assertion. To formulate this assertion we introduce the definition of sequences of type \(C(F_1,F_2)\).

First of all we introduce the definition of a limit function of a sequence. Let the sequence (1) satisfy the previous conditions. A function \(\varphi(x,E)\), defined almost everywhere on the set \(E\), \(\operatorname{mes} E>0\), \(E\subset [a,b]\), will be called a limit function of the sequence (1) if it is possible to define an increasing sequence of natural numbers \(n_\nu\) \((\nu=0,1,2,\ldots)\) such that

\[ \lim_{\nu\to\infty} f_{n_\nu}(x)=\varphi(x,E) \]

almost everywhere on \(E\).

Suppose further that \(M=\{\varphi(x,E)\}\) is some set of functions, each of which is defined almost everywhere on the corresponding set \(E\), \(\operatorname{mes} E>0\), \(E\subset [a,b]\), and take two measurable functions \(F_1(x)\) and \(F_2(x)\) satisfying inequality (3) almost everywhere on \([a,b]\). We shall say that the set \(M\) has the property \(h(F_1,F_2)\) if there exists a sequence of measurable functions, defined almost everywhere on \([a,b]\), for which \(M\) is the set of all limit functions, and \(F_1(x)\) and \(F_2(x)\) are respectively its lower and upper limits in measure on \([a,b]\).

A sequence of type \(C(F_1,F_2)\) will mean a sequence (1) of measurable functions, defined almost everywhere on \([a,b]\), which satisfies the following condition:

For any four measurable functions \(\varphi_1(x),\psi_1(x),\psi_2(x),\varphi_2(x)\), satisfying inequalities (5) almost everywhere on \([a,b]\), and for any set \(M=\{\varphi(x,E)\}\) possessing the property \(h(\psi_1,\psi_2)\), it is possible to define an increasing sequence of natural numbers \(m_k\) \((k=0,1,2,\ldots)\) such that \(M\) is the set of all limit functions of the sequence (6) and, moreover, the equalities \(\alpha^0\) and the condition \(\beta^0\) hold.

It is clear that every sequence of type \(C(F_1,F_2)\) is also a sequence of type \(B(F_1,F_2)\).

Theorem 1. The class of sequences of type \(A(F_1,F_2)\) coincides with the class of sequences of type \(C(F_1,F_2)\).

Hence, as a consequence, we obtain

Theorem 2. The class of sequences of type \(A(F_1,F_2)\) coincides with the class of sequences of type \(B(F_1,F_2)\).

The last theorem is a generalization of the above-mentioned theorem on trigonometric series.

We shall now consider sequences that are universal with respect to some summability method.

Take a method \(T\), defined by the matrix \(\|a_{mk}\|\) \((m,k=0,1,2,\ldots)\). We shall say that the sequence (1) of measurable functions, finite almost everywhere on some segment \([a,b]\), is a universal sequence with respect to the method \(T\), if for every measurable function, defined almost everywhere on \([a,b]\), one can choose such an increasing sequence of natural numbers

numbers \(m_k\) \((k=0,1,2,\ldots)\), such that the sequence (6) is summable by the method \(T\) to \(f(x)\) almost everywhere on \([a,b]\).

If \(T\) is a completely regular method*, then it is obvious that any sequence satisfying the preceding conditions and universal in the ordinary sense will also be universal with respect to the method \(T\).

It turns out that the latter assertion can be extended to methods \(T\) that transform every sequence converging to zero into a sequence converging to zero, if one additionally requires that these methods be normal, i.e., such that

\[ a_{mk}=0\quad (k>m), \qquad a_{mm}\ne 0\quad (m=0,1,2,\ldots), \]

where \(a_{mk}\) are the elements of the matrix defining the method \(T\) under consideration.

Received
13 V 1963

REFERENCES

\(^{1}\) D. Menshov, Uchen. zap. MGU, 7, issue 165, Mathematics, 3 (1954).
\(^{2}\) D. Menshov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 32, 3 (1950).
\(^{3}\) G. Hardy, Divergent Series, IL, 1951.

* The definition of a completely regular method is given in (3) (1,4, p. 24).