  Given a $\sigma$-finite infinite measure space $(\Omega,\mu)$, it is shown
that any Dunford-Schwartz operator $T:\,\mathcal L^1(\Omega)\to\mathcal
L^1(\Omega)$ can be uniquely extended to the space $\mathcal
L^1(\Omega)+\mathcal L^\infty(\Omega)$. This allows to find the largest
subspace $\mathcal R_\mu$ of $\mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$
such that the ergodic averages $\frac1n\sum\limits_{k=0}^{n-1}T^k(f)$ converge
almost uniformly (in Egorov's sense) for every $f\in\mathcal R_\mu$ and every
Dunford-Schwartz operator $T$. Utilizing this result, almost uniform
convergence of the averages $\frac1n\sum\limits_{k=0}^{n-1}\beta_kT^k(f)$ for
every $f\in\mathcal R_\mu$, any Dunford-Schwartz operator $T$ and any bounded
Besicovitch sequence $\{\beta_k\}$ is established. Further, given a measure
preserving transformation $\tau:\Omega\to\Omega$, Assani's extension of
Bourgain's Return Times theorem to $\sigma$-finite measure is employed to show
that for each $f\in\mathcal R_\mu$ there exists a set $\Omega_f\subset\Omega$
such that $\mu(\Omega\setminus\Omega_f)=0$ and the averages
$\frac1n\sum\limits_{k=0}^{n-1}\beta_kf(\tau^k\omega)$ converge for all
$\omega\in\Omega_f$ and any bounded Besicovitch sequence $\{\beta_k\}$.
Applications to fully symmetric subspaces $E\subset\mathcal R_\mu$ are given.
