  Consider an intersection measure $\ell_t ^{\mathrm{IS}}$ of $p$ independent
(possibly different) $m$-symmetric Hunt processes up to time $t$ in a metric
measure space $E$ with a Radon measure $m$. We derive a Donsker-Varadhan type
large deviation principle for the normalized intersection measure $t^{-p}\ell_t
^{\mathrm{IS}}$ on the set of finite measures on $E$ as $t \rightarrow \infty$,
under the condition that $t$ is smaller than life times of all processes. This
extends earlier work by W. K\"onig and C. Mukherjee (2013), in which the large
deviation principle was established for the intersection measure of $p$
independent $N$-dimensional Brownian motions before exiting some bounded open
set $D \subset \mathbb{R}^N$. We also obtain the asymptotic behaviour of
logarithmic moment generating function, which is related to the results of X.
Chen and J. Rosen (2005) on the intersection measure of independent Brownian
motions or stable processes. Our results rely on assumptions about the heat
kernels and the 1-order resolvents of the processes, hence include rich
examples. For example, the assumptions hold for $p\in \mathbb{Z}$ with $2\leq p
< p_*$ when the processes enjoy (sub-)Gaussian type or jump type heat kernel
estimates, where $p_*$ is determined by the Hausdorff dimension of $E$ and the
so-called walk dimensions of the processes.
