Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of $\kappa$ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves $\kappa^{<\kappa}$ and $\kappa^+$, but not $P(\kappa)$, destroys the measurability of $\kappa$, even if $\kappa$ is initially supercompact, strong, or if $I_1(\kappa)$ holds. Obtained as an application of some general lifting theorems, this result is an "inner model" type of theorem proved instead by forcing.