A note on the von Weizs\"{a}cker theorem

The von Weizs"{a}cker theorem states that every sequence of nonnegative random variables has a subsequence which is Ces`{a}ro convergent to a nonnegative random variable which might be infinite. The goal of this note is to provide a description of the set where the limit is finite. For this purpose, we use a decomposition result due to Brannath and Schachermayer.