This is an announcement for the paper "Forward-convex convergence of sequences in $\mathbb{L}^0_+$" by Constantinos Kardaras and Gordan Zitkovic. Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance. Archive classification: math.FA math.PR Mathematics Subject Classification: 46A16; 46E30; 60A10 Remarks: 14 pages The source file(s), fcc.bbl: 3371 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1002.1889 or http://arXiv.org/abs/1002.1889 or by email in unzipped form by transmitting an empty message with subject line uget 1002.1889 or in gzipped form by using subject line get 1002.1889 to: math@arXiv.org.