This is an announcement for the paper "Kolmogorov compactness criterion in variable exponent Lebesgue spaces" by Humberto Rafeiro. Abstract: The well-known Kolmogorov compactness criterion is extended to the case of variable exponent Lebesgue spaces $L^{p(\cdot)}(\overline{\Omega})$, where $\Omega$ is a bounded open set in $\mathbb R^n$ and $p(\cdot)$ satisfies some ``standard'' conditions. Our final result should be called Kolmogorov-Tulajkov Sudakov compactness criterion, since it includes the case $p_-=1$ and requires only the ``uniform'' condition. Archive classification: math.FA Mathematics Subject Classification: 46B50, 46E30 Remarks: 8 pages The source file(s), kolmogorov_18_03_2009.tex: 23807 bytes, is(are) stored in gzipped form as 0903.3214.gz with size 8kb. The corresponding postcript file has gzipped size 76kb. Submitted from: hrafeiro@ualg.pt The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0903.3214 or http://arXiv.org/abs/0903.3214 or by email in unzipped form by transmitting an empty message with subject line uget 0903.3214 or in gzipped form by using subject line get 0903.3214 to: math@arXiv.org.