On Riemann sums and maximal functions in $\mathbb R^n$

We investigate problems on a.e. convergence of Riemann sums
$$\begin{equation*} R_nf(x)=\frac1n\sum_{k=0}^{n-1}f\biggl(x+\frac kn\biggr), \qquad x\in\mathbb T, \end{equation*}$$
with the use of classical maximal functions in $\mathbb R^n$. A theorem on the equivalence of Riemann and ordinary maximal functions is proved, which allows us to use techniques and results of the theory of differentiation of integrals in $\mathbb R^n$ in these problems. Using this method we prove that for a certain sequence $\{n_k\}$ the Riemann sums $R_{n_k}f(x)$ converge a.e. to $f\in L^p$, $p>1$.
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