  In this paper we investigate problems on almost everywhere convergence of
subsequences of Riemann sums \md0
R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT.
\emd We establish a relevant connection between Riemann and ordinary maximal
functions, which allows to use techniques and results of the theory of
differentiations of integrals in $\ZR^n$ in mentioned problems. In particular,
we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums
$R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.

Classical Analysis and ODEs