A Tranversality Theorem for Holomorphic Mappings and Stability of Eisenman-Kobayashi Measures

We show that Thom's Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping $f:S\rightarrow M$ from a Stein manifold $S$ to a complex manifold $M$ and given an analytic subset $A$ of the jet space $J^{k} (S, M), \; f$ can be approximated in neighborhoods of compacts by holomorphic mappings whose $k$-jet extensions are transversal to $A$. As an application the stability of Eisenman-Kobayshi intrinsic $k$-measures with respect to deleting analytic subsets of codimension $>k$ is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.