# Anti Arzela-Ascoli

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?

• In theorem 1 you mean "a bounded sequence" don't you? Otherwise you may take a sequence of constant functions $f_n=n$, so it's even true with "every x" – Pietro Majer Apr 2 at 14:27
• Right, sorry I forgot to include that. Will edit the post, thanks! – Nate River Apr 2 at 14:31
• I would state the complete assumptions, in the question as well – Pietro Majer Apr 2 at 14:39
• Yep, I’ve added the equibounded condition to the question statement. Is there anything else I should add? – Nate River Apr 2 at 14:40
• I like the first one. Edited. – Nate River Apr 5 at 13:08

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