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 Cesaro sense. To illustrate, we consider the following two examples. The first of these was proposed by Yuval Peres in discussion on a seperate 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 Cesaro 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 Cesaro 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?
Best Answer
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, Acta Mathematica Academiae Scientiarum Hungaricae, 16 (1965), pp. 310–318.
[2] Janos Komlos, A generalization of a problem of Steinhaus, Acta Mathematica Academiae Scientiarum Hungaricae, 18 (1967), pp. 217–229.
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