A strong law of large numbers for positive random variables

Abstract

In the spirit of the famous Komlós (1967) theorem, every sequence of nonnegative, measurable functions ${\{f_n\}}_{n\in \mathbb{N}}$ on a probability space contains a subsequence which—along with all its subsequences—converges a.e. in Cesàro mean to some measurable $f_{\ast }:\mathrm{\Omega }\to [0,\mathrm{\infty }]$. This result of von Weizsäcker (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 Cesàro means.