Abstract
We describe a weak version of Laver indestructibility for a $\mu$-tall cardinal $\kappa$, $\mu > \kappa^{+}$, where ''weaker'' means that the indestructibility refers only to the Cohen forcing at $\kappa$ of a certain length. A special case of this construction is: if $\mu$ is equal to $\kappa^{+n}$ for some $1 < n < \omega$, then one can get a model $V^*$ where $\kappa$ is measurable, and its measurability is indestructible by $\Add(\kappa,\alpha)$ for any $0 < \alpha<\kappa^{+n}$ .