Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A theorem of Besicovitch and a generalization of the Birkhoff Ergodic
  Theorem   A remarkable theorem of Besicovitch is that an integrable function $f$ on
$\mathbb{R}^2$ is strongly differentiable if and only if its associated strong
maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's
result in the context of ergodic theory that provides a generalization of
Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable
function on a standard probability space and $T$ is an invertible
measure-preserving transformation on that space, then the ergodic averages of
$f$ with respect to $T$ converge a.e. if and only if the associated ergodic
maximal function $T^*f$ is finite a.e.

9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th