A Strong Law of Large Numbers for Positive Random Variables

In the spirit of the famous KOMLÓS (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 CESÀRO mean to some measurable $f_* : Ω\to [0, \infty]$. This result of VON WEIZSÄCKER (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 CESÀRO means.

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