  In this paper, we prove that stochastic porous media equations over
$\sigma$-finite measure spaces $(E,\mathcal{B},\mu)$, driven by time-dependent
multiplicative noise, with the Laplacian replaced by a self-adjoint transient
Dirichlet operator $L$ and the diffusivity function given by a maximal monotone
multi-valued function $\Psi$ of polynomial growth, have a unique solution. This
generalizes previous results in that we work on general measurable state
spaces, allow non-continuous monotone functions $\Psi$, for which, no further
assumptions (as e.g. coercivity) are needed, but only that their multi-valued
extensions are maximal monotone and of at most polynomial growth. Furthermore,
an $L^p(\mu)$-It\^{o} formula in expectation is proved, which is not only
crucial for the proof of our main result, but also of independent interest. The
result in particular applies to fast diffusion stochastic porous media
equations (in particular SOC models) and cases where $E$ is a manifold or a
fractal, and to non-local operators $L$, as e.g. $L=-f(-\Delta)$, where $f$ is
Bernstein function.
