Convergence of sequences of Riemann sums

We consider Riemann sums of the form
$$ M_n(f,x)=\frac1n\sum_{k=0}^{n-1}f\Bigl(x+\frac kn\Bigr);\quad R_n(f,x)\frac1n\sum_{k=0}^{n-1}f\Bigl(\frac{x+k}n\Bigr) $$
for measurable functions with period 1. We answer in the affirmative the question concerning the possibility of convergence almost everywhere on $(0,1)$ of these and other sums to different limits along different subsequences. For functions monotonic on the interval $(0,1)$ we investigate how slowly the sequences of subscripts can increase along which the convergence to different limits takes place [in the sense of convergence for all $x\in(0,1)$ for the sums $R_n(f,x)$ and in the sense of convergence in measure on $(0,1)$ for the sums $M_n(f,x)$].