Spectral measure of an eigenvector

Let $$T$$ be an unbounded selfadjoint operator and let $$P_T$$ denote it's spectral measure such that $$T= \int_\mathbb{R}\lambda dP_T (\lambda)$$. Suppose $$\psi$$ is an eigenvector of $$T$$ such that $$T\psi=\lambda \psi$$ for $$\lambda \in \mathbb{R}$$. I want to compute the corresponding spectral measure $$\mu_\psi(\Omega):=\langle \psi|P_T(\Omega)\psi \rangle$$. Obviously it has to be $$\mu_\psi=\delta_\lambda$$, where $$\delta_\lambda$$ denotes the Dirac measure centered at $$\lambda$$, when $$||\psi||=1$$. But I've so far failed to prove that. How do you do that? I don't want to use any explicit formula for $$P_T$$, and want to only use the property $$T= \int_\mathbb{R}\lambda dP_T (\lambda)$$.