On Riemann sums and maximal functions in $ZR^n$

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}fbigg(x+frac{k}{n}bigg),quad xin 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 $fin L^p$ with $p>1$.