The Lévy-Solovay theorem says that small forcings do not create measures.  J.D. Hamkins has generalized this to a larger class of forcings called gap forcings.  I would assume this cannot be generalized to _all_ forcings, but I cannot think of a counterexample.
Is there a forcing notion that creates a $\kappa$-complete (or even countably complete) measure $\mu$ on some uncountable cardinal $\kappa$ such that $\mu \cap V$ is not in $V$?