This is an announcement for the paper "An isomorphic version of the Busemann-Petty problem for arbitrary measures" by Alexander Koldobsky and Artem Zvavitch. Abstract: We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then $\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants for some special classes of measures and bodies. Finally, we prove a version of the hyperplane inequality for convex measures. Archive classification: math.FA math.MG Mathematics Subject Classification: 52A20 Submitted from: koldobskiya@missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1405.0567 or http://arXiv.org/abs/1405.0567