A note on the von Weizsäcker theorem
Stefan Tappe
TL;DR
This note sharpens the von Weizsäcker theorem by characterizing the finite-limit set of Cesàro-averaged subsequences of nonnegative $L^0$ random variables. It leverages the Brannath–Schachermayer convex-analytic decomposition to separate a finite-bounded region from an infinite-limit region, showing $\{ \xi<\infty \} = \Omega_b = \bar{\Omega}_b$ and $\{ \xi=\infty \} = \Omega_u = \bar{\Omega}_u$ (up to null sets), and links finiteness to the existence of an equivalent measure $\mathbb{Q} \approx \mathbb{P}$ under which the finite-part is $L^1(\mathbb{Q})$-bounded. The paper also provides an atom-wise description on atomic spaces and a dichotomy under SLLN: finite $\mu$ yields bounded convex hulls, while infinite $\mu$ yields hereditary unboundedness, with several natural conditions guaranteeing SLLN. Collectively, these results connect Cesàro limit finiteness to probabilistic tightness, measure changes, and weak convergence of convex combinations, refining the original theorem and offering practical criteria for when the limit is almost surely finite.
Abstract
The von Weizsäcker theorem states that every sequence of nonnegative random variables has a subsequence which is Cesàro 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.