  Let $X$ be a non-compact symmetric space of dimension $n$. We prove that if
$f\in L^{p}(X)$, $1\leq p\leq 2$, then the Riesz means $S_{R}^{z}\left(
f\right)$ converge to $f$ almost everywhere as $R\rightarrow \infty $, whenever
$\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) $.
