u_{s}\right\|_2
^{2}+\int_{0}^{T}E\, \mathcal{E}\left( u_{s}\right) ds\right)
^{1/2}\;< \; \infty \;.$$ We define ${{\cal H}}_{loc}={{\cal H}}_{loc}(\mathcal{O})$ to be the set of $H^1_{loc} ({{\cal O}})$-valued predictable processes defined on $[0,T]$ such that for any compact subset $K$ in ${{\cal O}}$:

$$\left( E \sup_{0\leq s\leq T}\int_K u_{s}(x)^{2}\, dx
+E\int_{0}^{T}\int_K |\nabla u_{s}(x)|^2\, dx ds\right) ^{1/2}\;< \;
\infty .$$

The space of test functions is the algebraic tensor product $\mathcal{D}=\mathcal{C} _{c}^{\infty }({{\ensuremath{\mathbb R}} }^+)\otimes
\mathcal{C}_c^2 ({\ensuremath{\mathcal O}})$, where $\mathcal{C} _{c}^{\infty }({{\ensuremath{\mathbb R}} }^+)$ denotes the space of all real infinite differentiable functions with compact support in $\mathbb{R}^+$ and $\mathcal{C}_c^2 ({\ensuremath{\mathcal O}})$ the set of $C^2$-functions with compact support in ${\ensuremath{\mathcal O}}$.

Now we recall the definition of the regular measure which has been defined in [@DMZ12].\
$\mathcal{K}$ denotes $L^\infty([0,T];L^2(\mathcal{O}))\cap
L^2([0,T];H_0^1(\mathcal{O}))$ equipped with the norm: $$\begin{aligned}
\parallel
v\parallel^2_\mathcal{K}&=&\parallel
v\parallel^2_{L^\infty([0,T];L^2(\mathcal{O}))}+\parallel
v\parallel^2_{L^2([0,T];H_0^1(\mathcal{O}))}\\
&=&\sup_{t\in[0,T[}\parallel v_t\parallel^2 +\int_0^T \left(
\parallel v_t \parallel^2  +\mathcal{E}(v_t)\right)\, dt
.\end{aligned}$$ $\mathcal{C}$ denotes the space of continuous functions with compact support in $[0,T[\times\mathcal{O}$ and finally: $$\mathcal {W}=\{\varphi\in L^2([0,T];H_0^1(\mathcal{O}));\ \frac{\partial\varphi}{\partial t}\in
L^2([0,T];H^{-1}(\mathcal{O}))\},$$ endowed with the norm$\parallel\varphi\parallel^2_{\mathcal {W}}=\parallel
\varphi\parallel^2_{L^2([0,T];H_0^1(\mathcal{O}))}+\parallel\displaystyle\frac{\partial
\varphi}{\partial t}\parallel^2_{L^2([0,T];H^{-1}(\mathcal{O}))}$.\
It is known (see [@LionsMagenes]) that $\mathcal{W}$ is continuously embedded in $C([0,T]; L^2 ({\ensuremath{\mathcal O}}))$, the set of $L^2 ({\ensuremath{\mathcal O}})$-valued continuous functions on $[0,T]$. So without ambiguity, we 