I am interested in convergence results that are true for Riemann-integrable functions but fail for Lebesgue-integrable functions. I know three of these, which happen to be closely related.

Convergence of Riemann sums of a Riemann-integrable function $f : [a,b] \rightarrow {\bf R}$.

Convergence of means of Riemann-integrable functions along equidistributed sequences (Weyl criterion).

The weak convergence of $P_n$ to $P$ implies $\int f dP_n \rightarrow \int f dP$ for all $P$-Riemann-integrable $f$ ($P$-Riemann integrability meaning that the set of discontinuities of $f$ is negligible wrt $P$).

Any others?