  The Stein-Tomas restriction theorem on Euclidean space says one can
meaningfully restrict $\hat{f}$ to the unit sphere of $\mathbb{R}^n$ provided
$f \in L^p(\mathbb{R}^n)$ with $1 < p < 2$. This result can be rewritten in
terms of the estimates for the spectral measure of Laplacian. Guillarmou,
Hassell and Sikora formulated a sufficient condition of the restriction
theorem, via spectral measure, on abstract metric measure spaces. But they only
proved the result in a special case. The present note aims to give a complete
proof. In the end, it will be applied to the restriction theorem on
asymptotically conic manifolds.
