MATHEMATICS

JAN LIPIŃSKI

ON THE CONVERGENCE OF A SEQUENCE OF CONTINUOUS FUNCTIONS TO INFINITY

(Presented by Academician P. S. Aleksandrov on 18 V 1961)

Let \(f_n(x)\) be a sequence of continuous functions defined on the space \(\mathscr{E}\) of real numbers. Let

\[ F_1=\{x:\lim_{n\to\infty} f_n(x)=+\infty\}; \tag{1} \]

\[ F_2=\{x:\lim_{n\to\infty} f_n(x)=-\infty\}. \tag{2} \]

Then, as is known (see \((^1)\), p. 259),

\[ F_1=\bigcap_{m=1}^{\infty}\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}\{x:f_n(x)\ge m\}; \qquad F_2=\bigcap_{m=1}^{\infty}\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}\{x:f_n(x)\le -m\}. \tag{3} \]

Since the sets \(\{x:f_n(x)\ge m\}\), \(\{x:f_n(x)\le -m\}\) are closed, it follows that

\[ F_1\in F_{\sigma\delta},\qquad F_2\in F_{\sigma\delta}. \tag{4} \]

It is obvious that

\[ F_1\cap F_2=0. \tag{5} \]

Let \(F\in F_{\sigma\delta}\). Hahn \((^2)\), and also Sierpiński \((^3)\), proved that there exists a sequence of continuous functions \(f_n(x)\) such that
\(F=\{x:\lim_{n\to\infty} f_n(x)=0\}\) (see also \((^1)\), pp. 261—262).

Put \(\varphi_n(x)=[\sup(n^{-1}, |f_n(x)|)]^{-1}\). Then we have \(F=\{x:\lim_{n\to\infty}\varphi_n(x)=+\infty\}\). Thus we see that the set of all those points at which a sequence of continuous functions \(f_n(x)\) converges to \(+\infty\) has type \(F_{\sigma\delta}\). Conversely, every set of type \(F_{\sigma\delta}\) is the set of convergence to \(+\infty\) of some sequence of continuous functions.

The question arises: if conditions (4) and (5) are satisfied, does there exist a sequence of continuous functions \(f_n(x)\) for which (1) and (2) hold? I. P. Kornfeld observed that the answer to this question is negative. Indeed, the sets

\[ F_1\subset E^{(1)}=\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}\{x:f_n(x)\ge 1\}, \]

\[ F_2\subset E^{(2)}=\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}\{x:f_n(x)\le -1\}. \]

It is obvious that \(E^{(1)}\in F_\sigma\), \(E^{(2)}\in F_\sigma\), and \(E^{(1)}\cap E^{(2)}=0\). Using the terminology of N. N. Luzin, we may conclude that the sets \(F_1\) and \(F_2\) must be separable by sets of type \(F_\sigma\). I. P. Kornfeld also found some sufficient conditions for the pair of sets \(F_1, F_2\), but they differ from the necessary ones.

In the present paper we prove that the indicated necessary conditions are sufficient, i.e. for any two sets \(F_1\) and \(F_2\) of type \(F_{\sigma\delta}\) and separable by \(F_\sigma\), there exists a sequence of continuous functions \(f_n(x)\) for which (1) and (2) hold.

First we shall formulate several lemmas on sets of types \(F_\sigma\) and \(F_{\sigma\delta}\); then we shall define the functions \(f_n(x)\) and give a brief outline of the proof of the assertion stated above.

Lemma 1. Let \(A \in F_\sigma\). Then there exist sequences of open sets \(P_n\) and closed bounded sets \(K_n\) such that

\[ K_n \subset K_{n+1},\quad K_n \subset P_n,\quad A=\bigcup_{n=1}^{\infty} K_n=\lim_{n\to\infty} P_n=\lim_{n\to\infty} \overline{P}_n . \]

This lemma is first proved for the case when \(A\) is a set of type \(F_\sigma\) and of the first category; then when \(A\) is an open set. In the second case one obtains the result

\[ A=\lim_{n\to\infty} P_n=\lim_{n\to\infty} \overline{P}_n \]

(more precise than

\[ A=\lim_{n\to\infty} P_n=\lim_{n\to\infty} \overline{P}_n). \]

Since an arbitrary set \(A\in F_\sigma\) is the sum of an open set and a set \(F_\sigma\) of the first category, from the validity of the lemma for the special cases one can prove its validity also in the general case.

Lemma 2. Let \(E^{(1)}\in F_\sigma\), \(E^{(2)}\in F_\sigma\), and \(E^{(1)}\cap E^{(2)}=0\). Then there exist open sets \(L_j^{(i)}\) \((i=1,2;\ j=1,2,\ldots)\) and closed bounded sets \(K_j^{(i)}\), satisfying the conditions

\[ \overline{L}_j^{(1)}\cap \overline{L}_j^{(2)}=0,\quad K_j^{(i)}\subset K_{j+1}^{(i)},\quad K_j^{(i)}\subset L_j^{(i)},\quad E^{(i)}=\bigcup_{j=1}^{\infty} K_j^{(i)}=\lim_{j\to\infty} L_j^{(i)}=\lim_{j\to\infty} \overline{L}_j^{(i)} . \]

Lemma 3. Let

\[ E_1=\bigcup_{i=1}^{\infty} F_i, \]

where \(F_i\subset F_{i+1}\) are closed bounded sets, and let \(G_i\) be open sets such that \(F_i\subset G_i\),

\[ E_1=\lim_{i\to\infty} G_i . \]

Then, if \(E_2\in F_\sigma\) and \(E_2\subset E_1\), there exists a sequence of open sets \(B_i\) and a sequence of closed bounded sets \(H_i\) such that

\[ H_i\subset H_{i+1},\quad H_i\subset B_i,\quad \overline{B}_i\subset G_i,\quad E_2=\bigcup_{i=1}^{\infty} H_i =\lim_{i\to\infty} B_i=\lim_{i\to\infty} \overline{B}_i . \]

For the proof of Lemmas 2 and 3, Lemma 1 is used.

Lemma 4. Let \(F_i\in F_{\sigma\delta}\) \((i=1,2)\), \(E^{(i)}\in F_\sigma\), \(F_i\subset E^{(i)}\), and \(E^{(1)}\cap E^{(2)}=0\). Then there exist open sets \(L_{j,k}^{(i)}\) \((j=1,2,\ldots;\ k=1,2,\ldots)\) such that

\[ \overline{L}_{1,k}^{(1)}\cap \overline{L}_{1,k}^{(2)}=0,\quad L_{j,k}^{(i)}\supset \overline{L}_{j+1,k}^{(i)}, \]

\[ F_i=\lim_{j\to\infty}\lim_{k\to\infty}\overline{L}_{j,k}^{(i)} . \tag{6} \]

For the proof of the lemma, first Lemma 2 is used, and then Lemma 3.

Theorem. Let for a pair of sets \(F_i\in F_{\sigma\delta}\) \((i=1,2)\) there exist a pair of sets \(E^{(i)}\in F_\sigma\), satisfying the conditions \(F_i\subset E^{(i)}\), \(E^{(1)}\cap E^{(2)}=0\). Then there exists a sequence of continuous functions \(f_n(x)\) such that equalities (1) and (2) are fulfilled.

Let the sets \(L_{j,k}^{(i)}\) \((i=1,2)\) have the properties indicated in Lemma 4. Then the sets

\[ \overline{L}_{n,n}^{(1)},\quad \overline{L}_{n,n}^{(2)},\quad \overline{L}_{j,n}^{(1)}\setminus L_{j,n}^{(1)},\quad \overline{L}_{j,n}^{(2)}\setminus L_{j,n}^{(2)} \quad (j=1,2,\ldots,n-1) \]

are closed and have no pairwise common points. Put

\[ f_n(x)= \begin{cases} (-1)^{i+1}n, & \text{for } x\in \overline{L}_{n,n}^{(i)},\\ (-1)^{i+1}j, & \text{for } x\in \overline{L}_{j,n}^{(i)}\setminus L_{j,n}^{(i)}, \end{cases} \]

where \(i=1,2;\ j=1,2,\ldots,n-1\). In each component interval of the open set

\[ \mathscr{E}\setminus\bigl(\overline{L}_{1,n}^{(1)}\cup \overline{L}_{1,n}^{(2)}\bigr), \]

whose boundary points belong to two different sets \(\overline{L}_{1,n}^{(1)}\) and \(\overline{L}_{1,n}^{(2)}\), the function \(f_n(x)\) is defined so that

so that in the closure of this component interval it is linear. In a component interval whose boundary points belong to the same set \(\overline L_{1,n}^{(i)}\), the function \(f_n(x)\) is defined by the equality
\[ f_n(x)=(-1)^{i+1}+(-1)^i \min [1/2,\rho(x,\overline L_{1,n}^{(i)})]. \]
Since at the boundary points of this component interval the function \(f_n(x)\) has the value \((-1)^{i+1}\), \(f_n(x)\) is continuous in the closure of the interval. Moreover, on this component interval the function \(f_n(x)\) satisfies the Lipschitz condition with constant 1. It remains now to define \(f_n(x)\) at the points of the sets \(L_{j-1,n}^{(i)}\setminus \overline L_{j,n}^{(i)}\) \((1<j\le n)\). These sets are open, and at the boundary points of their component intervals the function \(f_n(x)\) has already been defined. If the boundary points of an interval belong to two different sets \(\overline L_{j-1,n}^{(i)}\), \(\overline L_{j,n}^{(i)}\), or both belong to the set \(\overline L_{j-1,n}^{(i)}\), then \(f_n(x)\) is defined so that it is linear in the closure of the interval. If both boundary points belong to \(\overline L_{j,n}^{(i)}\), then let, in this component interval,
\[ f_n(x)=(-1)^{i+1}j+(-1)^i \min [1/2,\rho(x,\overline L_{j,n}^{(i)})]. \]
Thus, in the closure of an arbitrary component interval the function \(f_n(x)\) is either linear or satisfies the Lipschitz condition with constant 1.

It is possible to verify that all the functions \(f_n(x)\) are continuous and that
\[ E_{m,n}^{(1)}=\{x:f_n(x)\ge m\} = \begin{cases} \overline L_{m,n}^{(1)}, & \text{for } 1\le m\le n,\\ 0, & \text{for } m>n; \end{cases} \]
\[ E_{m,n}^{(2)}=\{x:f_n(x)\le -m\} = \begin{cases} \overline L_{m,n}^{(2)}, & \text{for } 1\le m\le n,\\ 0, & \text{for } m>n. \end{cases} \]
Hence it follows that
\[ \lim_{n\to\infty} E_{m,n}^{(i)}=\lim_{n\to\infty}\overline L_{m,n}^{(i)}. \]
On the basis of (3) and of the preceding,
\[ \{x:\lim_{n\to\infty} f_n(x)=+\infty\} = \bigcap_{m=1}^{\infty}\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}E_{m,n}^{(1)} = \lim_{m\to\infty}\lim_{n\to\infty}\overline L_{m,n}^{(1)}, \]
\[ \{x:\lim_{n\to\infty} f_n(x)=-\infty\} = \bigcap_{m=1}^{\infty}\bigcup_{j=1}^{\infty}\bigcap_{n=j}^{\infty}E_{m,n}^{(2)} = \lim_{m\to\infty}\lim_{n\to\infty}\overline L_{m,n}^{(2)}. \]
Taking (6) into account, we see that equalities (1) and (2) are valid for the constructed sequence of functions \(f_n(x)\), as was required to prove.

In conclusion the author expresses sincere gratitude to Prof. D. E. Menshov, under whose supervision this work was written, and also to Prof. P. L. Ulyanov for posing the problem and for a number of valuable suggestions.

Moscow State University
named after M. V. Lomonosov

Received
16 V 1961

CITED LITERATURE

1. F. Hausdorff, Set Theory, Moscow–Leningrad, 1937.
2. H. Hahn, Arch. d. Math. u. Phys., 28 (1919).
3. W. Sierpiński, Fund. Math., 2 (1921).