Upper density and a theorem of Banach

Abstract: Suppose $A_n$ $(n\in\mathbb{N})$ is a sequence of sets in a finitely-additive measure space which are uniformly bounded away from $0$, $\mu{A_n}\ge a>0$ for all $n$. Then there is a subsequence $A_{n_k}$, where $\{n_k\}_k$ has upper Banach density $\ge a$, such that $\mu\bigcap_{k

\textbf{Theorem:} Let $\{\,f_n : n\in\mathbb{N}\}$ be a uniformly bounded sequence of functions on a set $X$. The following are equivalent: (i)~$\{f_n\}_n$ weakly d-converges to $0$; (ii)~for any sequence $\{x_k : k\in\mathbb{N}\}$ in $X$, $d$-$\!\lim\limits_{n\to\infty}\liminf\limits_{k\to\infty}f_n(x_k)=0$.

Here ``d-" denotes a density limit. Banach's non-density version of this theorem (without the ``d-") has been described by some as ``marvelous".

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)