A strong law of large numbers for positive random variables

In the spirit of the famous Komlos (1967) theorem, every sequence of nonnegative, measurable functions {fn}n2N on a probability space contains a subsequence which- along with all its subsequences-converges a.e. in Cesaro mean to some measurable f* : S2 OE 0; oo]. This result of von Weizsacker (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of Delbaen and Schachermayer (1994), replacing general convex combinations by Cesaro means.