Mathematics

Yu. Flachsmeier (J. Flachsmeier)

On Dini Convergence in a Function Space

(Presented by Academician P. S. Aleksandrov on 27 IV 1963)

On compact Hausdorff spaces \(X\) there holds the well-known Dini convergence theorem, according to which an increasing or decreasing Moore–Smith sequence of continuous real-valued functions on \(X\), converging pointwise to a continuous function, converges uniformly to this limiting function (see \((^2)\), p. 53, Theorem 1). G. Hellmberg proved in \((^4)\) that this property characterizes compact spaces, and I. Glicksberg \((^3)\) proved that Dini’s theorem for ordinary sequences characterizes pseudocompact spaces. The investigations presented here are intended to supplement these results somewhat. For this purpose we define Dini convergence in a system of functions.

Definition 1. Let \(S(X)\) be a family of real-valued functions on the set \(X\). An element \(f \in S(X)\) is called a majorant (or minorant) of a Moore–Smith sequence \((f_i)_{i \in I}\) from \(S(X)\), if for almost all \(f_i\) with respect to the pointwise order one has \(f_i \leq f\) (respectively \(f \leq f_i\)) (i.e., if there exists an index \(i_0 \in I\) such that for all indices \(i > i_0\) following \(i_0\) and all \(x \in X\) one has \(f_i(x) \leq f(x)\) (respectively \(f(x) \leq f_i(x)\)).

Definition 2. Let \((f_i)_{i \in I}\) be a Moore–Smith sequence from \(S(X)\). Moore–Smith sequences \((g_j)_{j \in J}\), \((h_k)_{k \in K}\) from \(S(X)\) are called squeezing sequences for the sequence \((f_i)_{i \in I}\), if \((g_j)_{j \in J}\) is increasing, i.e. \(g_j \geq g_{j'}\) as soon as \(j > j'\), \((h_k)_{k \in K}\) is decreasing, i.e. \(h_k \leq h_{k'}\) as soon as \(k > k'\), and all \(g_j\) are minorants, while all \(h_k\) are majorants, of the sequence \((f_i)_{i \in I}\).

Definition 3. A Moore–Smith sequence \((f_i)_{i \in I}\) from \(S(X)\) will be called Dini-convergent in \(S(X)\) if for \((f_i)_{i \in I}\) there exist squeezing sequences in \(S(X)\) converging pointwise to one and the same function.

Remark. Let \((f_i)_{i \in I}\) be a given Dini-convergent sequence from \(S(X)\), and let \((g_j)_{j \in J}\), \((h_k)_{k \in K}\) be some squeezing sequences for it, converging pointwise to a function \(f\). Then the function \(f\) depends only on \((f_i)_{i \in I}\), and not on the squeezing sequences themselves \((g_j)_{j \in J}\), \((h_k)_{k \in K}\).

We shall call this function the Dini limit of the sequence \((f_i)_{i \in I}\), symbolically

\[ f_i \xrightarrow[D,S(X)]{} f. \]

Here the indication of \(S(X)\) is important, as will be shown in Example 2. Obviously, Dini convergence implies pointwise convergence

\[ f_i \xrightarrow[D,S(X)]{} f \Rightarrow f_i \xrightarrow[\pi,\, lm]{} f . \]

In the case of a system \(S(X)\) of continuous functions on some topological space \(X\), Dini convergence implies even uniform convergence at points.

Lemma 1. Let \(S(X) \subset \mathscr{C}(X,R)\) be a subsystem of the family of all continuous real-valued functions of the topological space \(X\). If

\[ f_i \xrightarrow[D,S(X)]{} f, \]

then \((f_i)_{i \in I}\) converges uniformly at every point

\[ f_i \xrightarrow[p.t.]{} f, \]

i.e. for ...

for each \(\varepsilon>0\) and \(x_0\in X\) there exists a neighborhood \(U_{x_0}(\varepsilon)\) of the point \(x_0\) such that \(|f_i(x)-f(x)|<\varepsilon\) for all \(x\in U_{x_0}(\varepsilon)\) and almost all \(i\in I\).

In particular, \(f\) is a continuous function, i.e. \(f\in \mathscr C(X,R)\).

Proof. Let \(x_0\in X\) and let \((g_j)_{j\in J}\), \((h_k)_{k\in K}\) be squeezing sequences for \((f_i)_{i\in I}\) from \(S(X)\) converging to the function \(f\). Then, for each fixed \(\varepsilon>0\), there exist indices \(k_0\in K\), \(j_0\in J\) such that

\[ h_{k_0}(x_0)-g_{j_0}(x_0)<\varepsilon/2. \tag{*} \]

Since \(h_{k_0}\) and \(g_{j_0}\) are continuous, one can find neighborhoods \(V\) and \(W\) of the point \(x_0\) such that
\[ g_{j_0}(V)\subset \bigl(g_{j_0}(x_0)-\varepsilon/2,\; g_{j_0}(x_0)+\varepsilon/2\bigr) \]
and
\[ h_{k_0}(W)\subset \bigl(h_{k_0}(x_0)-\varepsilon/2,\; h_{k_0}(x_0)+\varepsilon/2\bigr). \]
For the neighborhood \(U=V\cap W\) of the point \(x_0\) we have
\[ g_{j_0}(y)\in \bigl(g_{j_0}(x_0)-\varepsilon/2,\; g_{j_0}(x_0)+\varepsilon/2\bigr) \]
and
\[ h_{k_0}(y)\in \bigl(h_{k_0}(x_0)-\varepsilon/2,\; h_{k_0}(x_0)+\varepsilon/2\bigr) \]
for \(y\in U\). From \((*)\) it follows, in view of the relation
\[ g_{j_0}(x_0)\leq f(x_0)\leq h_{k_0}(x_0), \]
that
\[ f(x_0)-g_{j_0}(x_0)<\varepsilon/2,\qquad h_{k_0}(x_0)-f(x_0)<\varepsilon/2. \]
Finally,
\[ f(x_0)-\varepsilon<g_{j_0}(x_0)-\varepsilon/2<g_{j_0}(y)\leq f_i(y)\leq h_{k_0}(y)< \]
\[ < h_{k_0}(x_0)+\varepsilon/2<f(x_0)+\varepsilon \]
for all \(y\in U\) and almost all \(i\in I\), i.e.
\[ f_i(U)\subset \bigl(f(x_0)-\varepsilon,\; f(x_0)+\varepsilon\bigr) \]
for almost all \(i\in I\).

In an analogous way we obtain
\[ f(U)\subset \bigl(f(x_0)-\varepsilon,\; f(x_0)+\varepsilon\bigr), \]
which completes the proof.

Example 1. Let \(X=N\) be the discrete space of natural numbers. We order the set of indices \(I=N\times N\) lexicographically, i.e.
\[ (n,m)\succ (n',m')\Longleftrightarrow n>n'\ \text{or}\ (n=n'\ \text{and}\ m>m'). \]

The Moore–Smith sequence over \(I\)
\[ f_{(n,m)}(x)= \begin{cases} m, & \text{for } x=n,\\ 0, & \text{for } x\ne n \end{cases} \]
converges uniformly at each point to the zero function \(f\equiv 0\), although it does not converge to it in the sense of Dini convergence (since \((f_{(n,m)})_{(n,m)\in N\times N}\) has no majorant in \(\mathscr C(X,R)\)).

Example 2. For the same space \(X=N\) and the family \(\mathscr C^*(X,R)\) of all bounded functions on \(X\), for the sequence
\[ f_n(x)= \begin{cases} n, & \text{if } x=n,\\ 0, & \text{if } x\ne n \end{cases} \]
we have \(f_n \xrightarrow[p.t.]{} f\equiv 0\), but not
\[ f_n \xrightarrow[D,\,\mathscr C^*(X,R)]{} f, \]
although
\[ f \xrightarrow[D,\,\mathscr C(X,R)]{} f; \]
the squeezing sequences are
\[ g_n(x)\equiv 0,\qquad h_n(x)= \begin{cases} m, & \text{if } x=m,\ \text{if } x\leq n,\\ 0, & \text{if } x>n. \end{cases} \]

Since for a compact space \(X\), from convergence uniform at each point there follows uniform convergence, and, on the other hand, from uniform convergence \(f_i\xrightarrow[p]{} f\) there always follows Dini convergence in
\[ \mathscr C^*(X,R) \]
(for this it is only necessary to consider the squeezing sequences
\[ f-\frac1n\quad \text{and}\quad f+\frac1n \]
), we may formulate the quoted Dini theorem symmetrically as follows.

Dini’s Theorem. For a compact Hausdorff space \(X\), Dini convergence and uniform convergence in the system of all real continuous functions on \(X\) are equivalent.

We can obtain the Helly converse of Dini’s theorem from the following result on the topologization of Dini convergence.

Theorem 1. Let \(X\) be a completely regular space. Dini convergence in the family \(\mathcal C^*(X,R)\) of all continuous real-valued bounded functions on \(X\) is a topological convergence (i.e., there exists a topology \(\tau\) in \(\mathcal C^*(X,R)\) such that \(f_i \xrightarrow[D,\mathcal C^*(X,R)]{} f\) and \(f_i \xrightarrow{\tau} f\) for any Moore–Smith sequences are equivalent) if and only if \(X\) is compact.

Corollary. (Helmberg \((^4)\)). If, for a completely regular space, Dini convergence and uniform convergence of Moore–Smith sequences in \(\mathcal C^*(X,R)\) are equivalent, then \(X\) is compact.

Proof of Theorem 1. The sufficiency of the compactness condition for \(X\) follows from the validity of Dini’s theorem. To prove its necessity, suppose that \(X\) is noncompact. Let \(\beta X\) be the Stone–Čech compactification of the space \(X\), and let \(x_0 \in \beta X \setminus X\). Denote by \(\mathfrak U\) the system of all closed neighborhoods of the point \(x_0\) in \(\beta X\). For each neighborhood \(U \in \mathfrak U\) and \(y \in X\), \(y \notin U\), and for an arbitrary natural number \(\nu \in N\), choose a continuous function \(\tilde f^\nu_{(U,y)}\) on \(\beta X\), equal to \(\nu\) on the set \(U\), taking the value \(0\) at the point \(y\), and intermediate values at the remaining points. By \(f^\nu_{(U,y)}\) denote the restriction of the function \(\tilde f^\nu_{(U,y)}\) to \(X\). Consider the following sets of indices:
\[ I(\nu)=\{i\mid i=(U_1,\ldots,U_n;y_1,\ldots,y_n);\quad U_k\in\mathfrak U,\ y_k\in X,\ \text{and } y_k\notin U_k\}. \]
On \(I(\nu)\) define the direction:
\[ i=(U_1,\ldots,U_n;y_1,\ldots,y_n)\succ j=(V_1,\ldots,V_m;z_1,\ldots,z_m) \]
\[ \Longleftrightarrow\ \text{for }(V_k,z_k)\text{ from }j\text{ there exist such }(U_l,y_l)\text{ from }i\text{ that }(V_k,z_k)=(U_l,y_l). \]
Over the sets \(I(\nu)\), \(\nu\in N\), define the following Moore–Smith sequences:
\[ f_i^\nu=f^\nu_{(U_1,\ldots,U_n;y_1,\ldots,y_n)} =\inf\bigl(f^\nu_{(U_k,y_k)}\bigr)\mid (U_k,y_k)\in i . \]
Then for each sequence \(\bigl(f_i^\nu\bigr)_{i\in I(\nu)}\) one has \(f_i^\nu\xrightarrow[D,\mathcal C^*(X,R)]{}f\equiv0\), because \(\bigl(f_i^\nu\bigr)_{i\in I(\nu)}\) is a decreasing sequence converging pointwise to \(f\equiv0\). If Dini convergence were topological, then the theorem on iterated limits would have to hold (see \((^6)\), p. 69). In the present case the sequence \(f_{(\nu,\mathfrak I)}=f^\nu_{\mathfrak I(\nu)}\); \(\mathfrak I\in\prod I(\nu)\mid \nu\in N\) \((\mathfrak I(\nu)=i_\nu,\ \text{if } \mathfrak I=(i_1,i_2,\ldots,i_\nu,\ldots))\) over the index set \(N\times\prod I(\nu)\mid \nu\in N\) with the order
\[ (\nu,\mathfrak I)\succ(\mu,\mathfrak I')\Longleftrightarrow \nu>\mu\ \text{and }\mathfrak I\succ\mathfrak I' \text{ coordinatewise}, \]
would have to be Dini-convergent in \(\mathcal C^*(X,R)\) to \(f\). But this is impossible, since for \(f_{(\nu,\mathfrak I)}\) in \(\mathcal C^*(X,R)\) there exists no majorant whatsoever.

Remark. Let \(\mathcal C_K(X,R)\) be the family of all real-valued continuous functions with compact supports (i.e., for every \(f\in\mathcal C_K(X,R)\) there exists a compact set \(K_f\) such that \(f(x)=0\) outside \(K_f\)).

Further, let \(\mathcal C_\infty(X,R)\) be the family of all real-valued continuous functions for which the sets \(\{x\mid |f(x)|\ge\varepsilon>0\}\) are compact for every fixed \(\varepsilon>0\).

Then the following hold:
\[ \text{1) }\quad f_i\xrightarrow[D,\mathcal C_K(X,R)]{} f\Rightarrow f\in\mathcal C_K(X,R). \]
\[ \text{2) }\quad f_i\xrightarrow[D,\mathcal C_\infty(X,R)]{} f\Rightarrow f\in\mathcal C_\infty(X,R). \]
\[ \text{3) }\quad f_i\xrightarrow[D,\mathcal C_K(X,R)]{} f\Rightarrow f_i\xrightarrow{p}f. \]
The converse is true only for a compact space \(X\).

\[ \text{4) If }X\text{ is locally compact, then} \]
\[ f_i\xrightarrow[D,\mathcal C_\infty(X,R)]{} f\Rightarrow f_i\xrightarrow{p}f . \]

(True not only for a locally compact space; the converse is valid only for a compact space \(X\).)

We shall now consider Dini convergence of ordinary sequences in \(\mathscr{C}^{*}(X,R)\) and, in addition to Theorem 1, prove the metrizability of this convergence.

Theorem 2. Let \(X\) be a completely regular space. Dini convergence of ordinary sequences in the family \(\mathscr{C}^{*}(X,R)\) of all continuous real bounded functions on \(X\) is metrizable (i.e., there exists a metric \(\rho\) in \(\mathscr{C}^{*}(X,R)\) such that \(f_n \xrightarrow[D,\mathscr{C}^{*}(X,R)]{} f\) and \(\rho(f_n,f)\to 0\) are equivalent for arbitrary ordinary sequences in \(\mathscr{C}^{*}(X,R)\)) if and only if \(X\) is pseudocompact (i.e., every real continuous function on \(X\) is bounded).

Proof. Iseki \({}^{5}\) and Bagley, Connell, McKnight \({}^{1}\) proved that, for pseudocompact spaces, from pointwise convergence of an ordinary sequence there follows, in general, uniform convergence of this sequence. By Lemma 1 this implies the equivalence of \(f_n \xrightarrow[D\mathscr{C}^{*}(X,R)]{} f\) and \(f_n \to f\) for ordinary sequences from \(\mathscr{C}^{*}(X,R)\). Since uniform convergence in \(\mathscr{C}^{*}(X,R)\) is metrizable, the first part is proved.

Now suppose that \(X\) is not pseudocompact. Then there exists in \(X\) a countable locally finite family of open sets \(V_1,V_2,\ldots\) with pairwise disjoint closures \({}^{3}\). In each set \(V_n\) choose a point \(x_n\) and fix functions with values between \(0\) and \(\nu\) such that

\[ g_m^\nu(x)= \begin{cases} 0, & x\notin \overline{V}_m,\\ \nu, & x=x_m. \end{cases} \]

Then

\[ f_n^\nu(x)=\sum_{m=n}^{\infty} g_m^\nu(x) \]

is a continuous function on \(X\); moreover,

\[ 0\leq f_n^\nu(x)\leq \nu,\qquad f_n^\nu(x)= \begin{cases} 0 & \text{for } x\notin \bigcup \overline{V}_m \mid m\geq n,\\ \nu & \text{for } x\in \{x_m\mid m\geq n\}. \end{cases} \]

For each fixed \(\nu\in N\) there is an ordinary decreasing sequence \((f_n^\nu)_{n\in N}\) Dini-convergent to \(f\equiv 0\).

If Dini convergence is metrizable, then one can choose indices

\[ n_1 n_2\leq \cdots \leq n_\nu\leq \cdots \]

so that \(f_{n_1}^{1}, f_{n_2}^{2},\ldots, f_{n_\nu}^{\nu},\ldots\) is Dini-convergent to \(f\equiv 0\). But this is impossible, because for this sequence in \(\mathscr{C}^{*}(X,R)\) there exists no majorant.

Corollary (Glicksberg \({}^{3}\), Iseki \({}^{5}\)). Let \(X\) be a completely regular space. \(X\) is pseudocompact if and only if Dini convergence and uniform convergence of ordinary sequences in \(\mathscr{C}^{*}(X,R)\) are equivalent.

Received
4 IV 1963

References

\({}^{1}\) R. W. Bagley, E. H. Connell, J. D. McKnight Jr., Proc. Am. Math. Soc., 9, 500 (1958).
\({}^{2}\) N. Bourbaki, Éléments de Mathématique, Livre III, chap. X, Paris, 1961.
\({}^{3}\) I. Glicksberg, Duke Math. J., 19, 253 (1952).
\({}^{4}\) G. Helmberg, Koninkl. Nederl. Akad. van Wetensch. Proc., Ser. A, 62, 419 (1959).
\({}^{5}\) K. Iséki, Proc. Japan Acad., 33, 320 (1957).
\({}^{6}\) J. L. Kelley, General Topology, 1955.