**Notation:** We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence $f\_n$ of real valued functions on $[0, 1] 
$ are equibounded if $\sup\_{n \in \mathbb N}\sup\_{x \in [0, 1]} |f\_n (x)| < \infty$.


**Some motivation:**


The Arzela Ascoli theorem for $C[0, 1]$ says that if we have an equibounded, equicontinuous sequence of functions, we have uniform convergence along a subsequence. What happens if we drop equicontinuity (but retain continuity of the functions)? What kind of convergence can we expect?


For any countable subset $S$ of $[0, 1]$, we can still diagonalize to get pointwise convergence on $S$ along a subsequence $f\_{n\_k}$ that depends on $S$. But conjecturally this is the best we can do in general. Indeed as far as pointwise convergence is concerned, we have:


**Theorem 1:** There exists an equibounded sequence of continuous functions $f\_n: [0, 1] \to \mathbb R$ such that for every increasing sequence $n\_k$ of naturals, $f\_{n\_k} (x)$ diverges for almost every $x \in [0, 1]$.


However, the examples that I am familiar with all rely on some sort of “independence” or equidistribution type argument. For such examples, one intuitively expects $f\_n$ to converge in the Cesàro sense. To illustrate, we consider the following two examples. The first of these was proposed by Yuval Peres in discussion on a separate forum.


**Example 1:** Take $f\_n (x) = \sin(nx)$.That $f\_n$ satisfy the conditions in Theorem 1 can be seen by noting that by Weyl’s criterion for equidistribution, the sequence $n\_{k}x \ \text{mod} \ 1$ is equidistributed for a.e. $x \in [0, 1]$. However by the same coin, we have that for *any* subsequence, $f\_{n\_k} (x)$ converges in the Cesàro sense for almost every $x$.


**Example 2:** Consider the domain $[0, 1]$ as a probability space, and take $g\_n$ to be the indicator function of independent events with probability $1/2$ each. Then an argument based on the second Borel-Cantelli lemma gives us that the $g\_n$ satisfy all conditions in Theorem 1 except continuity. We can then approximate the $g\_n$ by a sequence $f\_n$ of continuous functions, whence $f\_n$ satisfy the conditions in Theorem 1. But again it can be shown that for any subsequence, $f\_{n\_k} (x)$ converges in the Cesàro sense almost everywhere.


This suggests the following question:


**Question:** Does there exist an equibounded sequence of continuous functions $f\_n: [0, 1] \to \mathbb R$ such that for every increasing sequence $n\_k$ of naturals, $\lim\_{N \to \infty} \frac{1}{N} \sum\_{k = 0}^{N-1} f\_{n\_k}(x)$ almost everywhere fails to exist?



Anti Arzela-Ascoli
Under the stated conditions, there always exists a subsequence that Cesaro converges almost everywhere. This was a question of Steinhaus, solved by Revesz [1].
More generally, it suffices that the sequence $f\_n$ be uniformly bounded in $L^1$; This is a striking Theorem of Komlos [2] which in particular implies the Kolmogorov strong law of large numbers.


[1] P. Revesz, [On a problem of Steinhaus](https://doi.org/10.1007/BF01904839), Acta Mathematica Academiae Scientiarum Hungaricae, 16 (1965),
pp. 310–318.


[2] Janos Komlos, [A generalization of a problem of Steinhaus](https://doi.org/10.1007/BF02020976),
Acta Mathematica Academiae Scientiarum Hungaricae, 18 (1967), pp. 217–229.


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