Reports of the Academy of Sciences of the USSR

1. Vol. 152, No. 3

MATHEMATICS

S. G. KOZLOVTSЕV

ON THE STRUCTURE OF MEASURABLE FUNCTIONS DEVOID OF AN ASYMPTOTIC DERIVATIVE

(Presented by Academician A. N. Kolmogorov on 9 IV 1963)

In the present paper we study the structure of measurable functions having no asymptotic derivative. A. Ya. Khinchin \((^{1})\) established the following remarkable theorem:

Every function \(f(x)\), measurable and finite almost everywhere on a set \(E\), at almost every point \(x_0 \in E\) has one of the following two properties:
1) either it has a finite asymptotic derivative at the point \(x_0\);
2) or, for any two sets

\[ P'_{x_0}=P'_{x_0}(x\in E,\quad x>x_0),\qquad P''_{x_0}=P''_{x_0}(x\in E,\quad x<x_0) \]

of lower positive, respectively right and left, density at \(x_0\), the upper derivative numbers over these sets are equal to \(+\infty\), and the lower derivative numbers over the same sets are equal to \(-\infty\).

A. Ya. Khinchin’s theorem gives, to a certain extent, a characterization of the behavior of a function at points where the asymptotic derivative is absent (up to a set of points of measure zero). However, this characterization is too general. On the one hand, it is limited only to indicating the magnitudes of the limits

\[ \overline{\lim_{\substack{x\to x_0\\ x\in P_{x_0}}}} \frac{f(x)-f(x_0)}{x-x_0},\qquad \underline{\lim_{\substack{x\to x_0\\ x\in P_{x_0}}}} \frac{f(x)-f(x_0)}{x-x_0}, \]

where \(P_{x_0}\) is some (generally speaking, arbitrary) set of one-sided lower positive density at \(x_0\). On the other hand, A. Ya. Khinchin’s theorem says nothing about the density of the sets

\[ E_{a,b}(x_0)=\left\{x,\ a<\frac{f(x)-f(x_0)}{x-x_0}<b\right\} \tag{1} \]

in the case when \(a\) and \(b\) are finite and there is no asymptotic derivative at \(x_0\), except only that, for finite \(a\) and \(b\), the one-sided lower densities of the set \(E_{a,b}(x_0)\) at the point \(x_0\) cannot be positive.

For what follows, for any point \(x\) we shall denote by \(E_{s(\eta)}\) the set of all values \(x'\) for which the points \(\eta'=[x',f(x')]\) lie in the angle \(s(\eta)\) with vertex at the point \(\eta=[x,f(x)]\).

I. Ya. Plyamenov \((^{2})\) stated the following hypothesis:

Every measurable function \(f(x)\), finite almost everywhere, defined on a set \(E\), \(\operatorname{mes} E>0\), at almost every point \(x\in E\) has one of the following three properties: either 1) it has an asymptotic derivative, or 2) the upper density of the set \(E_{s(\eta)}\) at the point \(x\) is positive, whatever the angle \(s(\eta)\) with vertex at the point \(\eta=[x,f(x)]\), whose closure lies, excluding the vertex, in the half-plane \(x'>x\) (or \(x'<x\)), or else 3) the point \(x\) is a point of dispersion of the set \(E_{s(\eta)}\) for any angle \(s(\eta)\), \(\eta=[x,f(x)]\), satisfying the preceding conditions.

The validity of this hypothesis would mean, to a certain extent, the completion of the study of the structure of measurable functions by means of the sets \(E_{a,b}(x_0)\), defined by equality (1), for finite \(a\) and \(b\).

Let us introduce the following notation. Let \(x_0 \in E\); assuming that \(-\infty < a < b < +\infty\), denote by \(\alpha_{a,b}=\alpha_{a,b}(x_0)\) the upper right density of the set \(E_{a,b}(x_0)\) at the point \(x_0\). Similarly, denote by \(\beta_{a,b}=\beta_{a,b}(x_0)\) the upper left density of the set \(E_{a,b}(x_0)\) at the point \(x_0\).

Theorem 1. There exists a function \(f(x)\), measurable and finite on the segment \([0,1]\), such that for almost every \(x_0 \in [0,1]\) the following conditions are satisfied:

1) for any two numbers \(A\) and \(B\), \(0<A<B<\infty\),

\[ \alpha_{A,B}(x_0)=0; \]

2) for any two numbers \(A\) and \(B\), \(0<A<B<\infty\),

\[ \alpha_{-B,-A}(x_0)\ge \frac{B-A}{A} \]

(and, consequently, \(f(x)\) has no asymptotic derivative at \(x_0\));

3) there exists a set \(E_{x_0}^{(3)}\), having \(x_0\) as a point of upper right density equal to one, and such that: a) \(E_{x_0}^{(3)} \subset (x_0,1)\); b) if \(x \in E_{x_0}^{(3)}\), then \(f(x)<f(x_0)\); c) if \(x \in E_{x_0}^{(3)}\), then
\[ \frac{f(x)-f(x_0)}{x-x_0}\to 0 \]
as \(x\to x_0\).

Definition of the function \(f(x)\). In order to construct a function satisfying all the conditions of Theorem 1, we introduce the definition of an infinite \(A\)-fraction.

We shall call the half-interval \([0,1)\) a half-interval of rank zero and denote it by \(\Delta_i^0\), where \(i=1\). Divide the half-interval \([0,1)\) into \(4(=4^{2\cdot 0+1})\) equal non-overlapping half-intervals, each of length \(1/4\) (the left endpoint is included). Number all these half-intervals from left to right and call them half-intervals of the first rank. Denote the \(i\)-th half-interval of the first rank by \(\Delta_i^1\), \(i=1,2,3,4=4^{2\cdot 0+1}\). Each of the half-intervals of the first rank \(\Delta_i^1\) \((i=1,\ldots,4^{2\cdot 0+1})\) is divided into \(4^{2\cdot 1+1}\) equal non-overlapping half-intervals, each of length
\[ \frac{1}{4^{2^2}} \]
(the left endpoint is included). We shall call these half-intervals half-intervals of the second rank. Number from left to right all half-intervals of the second rank obtained by dividing one and the same half-interval of the first rank \(\Delta_i^1\), and denote them by \(\Delta_{i,j}^2\), where the index \(j\) \((j=1,2,\ldots,4^{2\cdot 1+1})\) determines the number of the half-interval \(\Delta_{i,j}^2\) in the system of all half-intervals of the second rank obtained by dividing the half-interval \(\Delta_i^1\) of the first rank. Finally, each of the half-intervals \(\Delta_{i,j}^2\) of the second rank is divided into \(4^{2\cdot 2+1}\) equal non-overlapping half-intervals of the third rank, each of length
\[ \frac{1}{4^{3^2}} \]
(as before, the left endpoint is included). Number all half-intervals of the third rank obtained by dividing one and the same half-interval \(\Delta_{i,j}^2\) from left to right, and denote them by \(\Delta_{j,k}^3\), where the index \(k\),

\[ k=1,\ldots,4^{2\cdot 2+1}, \]
determines the number of the given half-interval \(\Delta_{j,k}^3\) in the system of all half-intervals of the third rank obtained by dividing the given half-interval \(\Delta_{i,j}^2\) of the second rank*, and so on.

* In the notation \(\Delta_{j,k}^3\), the index \(i\) is not written.

Let now \(x \in (0,1)\); for any number \(n\), \(n=1,2,\ldots\), \(x\) belongs to one and only one of the half-intervals \(\Delta_{k,s}\) of rank \(n\). Put

\[ a_n = a_n(x)=s-1 \qquad (n=1,2,\ldots) \]

and we shall call the sequence \(0,a_1,a_2\ldots a_n\ldots\) the expansion of the given number \(x\) into an infinite \(A\)-fraction.

Define the function \(f(x)\) as follows:

1) if \(x \in (0,1)\) and \(0,a_1\ldots a_n\ldots\) is the expansion of \(x\) into an infinite \(A\)-fraction, then

\[ f(x)=\sum_{n=1}^{\infty}\frac{k_n(x)}{4^{\,n^2-n}}, \]

where

\[ k_n(x)=1,\qquad \text{if } a_n(x)\equiv 0 \pmod n, \]

\[ k_n(x)=0,\qquad \text{if } a_n(x)\not\equiv 0 \pmod n; \]

2) \(f(0)=f(1)=0\).

It turns out that the function \(f(x)\) so defined satisfies all the conditions of Theorem 1.

Thus the hypothesis put forward by I. Ya. Plamenevskii has proved to be false.

At the same time, Theorem 1, formulated in the present note, shows that a measurable function may have, generally speaking, a considerably more complicated structure than would follow from the hypothesis of I. Ya. Plamenevskii.

In the present note we shall record one more property of measurable functions. Let the function \(f(x)\) be defined on some set \(E\), and let \(a\) and \(b\) be arbitrary finite numbers, \(a<b\), and let \(\alpha_{a,b}(x_0)\) and \(\beta_{a,b}(x_0)\) be defined through the given set \(E_{a,b}(x_0)\) in the same way as before.

Define \(\alpha(x_0)\) and \(\beta(x_0)\) by the equalities

\[ \alpha(x_0)=\lim_{\substack{a\to-\infty\\ b\to+\infty}}\alpha_{a,b}(x_0),\qquad \beta(x_0)=\lim_{\substack{a\to-\infty\\ b\to+\infty}}\beta_{a,b}(x_0). \]

Theorem 2. Let the function \(f(x)\) be measurable and finite almost everywhere on the set \(E\), \(\operatorname{mes} E>0\).

Then

\[ \alpha(x)=\beta(x) \]

almost everywhere on \(E\).

Theorem 2 is easily obtained from a lemma of Hoang Tuy \((^3)\).

Received
9 IV 1963

References

\(^1\) D. Ya. Khinchin, Matem. sborn., 31, 285 (1924).
\(^2\) I. Ya. Plamenevskii, Matem. sborn., 42 (84), 2, 247 (1957).
\(^3\) Hoang Tuy, Matem. sborn., 54 (96), 2, 190 (1961).