Mean convergence for Banach space-valued random elements indexed in measure spaces

This article studies mean convergence of Banach space-valued random elements indexed in a family of finite measure spaces. We derive $L^p$-convergence theorems under (compact) uniform integrability in two regimes: a decaying-index-mass regime and a bounded-index-mass regime, the latter requiring a new dependence structure which is called diagonal negative dependence for the random elements and expressed via the self-product of the index measure. We provide examples showing that the conditions to obtain the results are sharp and strictly weaker than related conditions in the literature. As a further illustration for the index measure space framework, a functional law of large numbers in $L^p$ on the space of continuous functions is derived, where the random elements are solutions of stochastic differential equations driven by Brownian motions extracted from a common Brownian sheet.