This integral is understood as a Fourier transform of a temperate distribution and it is itself a temperate distribution.
From comments below:
1: How to understand $\int f(\tau) d\tau e(x,x,\tau)$: Since for Laplacian and elliptic operators in general $e(x,x,\tau)$ is a smooth function of $x, y$ with a value in the space of temperate distributions, there is no reason to worry how to understand this integral. Furthermore, it is not a measure, but more singular distribution in higher dimensions.
2: We do not recover $e(x,x,\tau)$ from $u(x,x,t)$ for all $t$ as in most cases it is impossible to construct an approximation for all $t$, but only for $t$ in small vicinity of $0$. Then we use Tauberian theorem of Hormander. I suggest to start from more elementary books; M. Shubin, Pseudodifferential operators and spectral theory would be the best.
