  Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset
L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable
functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient
conditions are given for almost uniform convergence in $X$ (in Egorov's sense)
of Ces\`aro averages $M_n(T)(f)=\frac1n\sum_{k = 0}^{n-1}T^k(f)$ for all
Dunford-Schwartz operators $T$ in $L^1(\Omega)+ L^\infty(\Omega)$ and any $f\in
X$. Besides, it is proved that the averages $M_n(T)$ converge strongly in $X$
for each Dunford-Schwartz operator $T$ in $L^1(\Omega)+L^\infty(\Omega)$ if and
only if $X$ has order continuous norm and $L^1(\Omega)$ is not contained in
$X$.
