  In the spirit of the famous KOML\'OS (1967) theorem, every sequence of
nonnegative, measurable functions $\{ f_n \}_{n \in \N}$ on a probability
space, contains a subsequence which - along with all its subsequences -
converges a.e. in CES\`ARO mean to some measurable $f_* : \Omega \to [0,
\infty]$. This result of VON WEIZS\"ACKER (2004) is proved here using a new
methodology and elementary tools; these sharpen also a theorem of DELBAEN &
SCHACHERMAYER (1994), replacing general convex combinations by CES\`ARO means.
