applies if for example $u$ is quasi-continuous.

To establish a maximum principle for local solutions we need to define the notion of [*local regular measures*]{}:

We say that a Radon measure $\nu$ on $[0,T[\times {\ensuremath{\mathcal O}}$ is a [local regular measure]{} if for any non-negative $\phi$ in $\mathcal{C}_c^\infty({\ensuremath{\mathcal O}})$, $\phi\nu$ is a regular measure.

Local regular measures do not charge polar sets (i.e. sets of capacity 0).

Let $A$ be a polar set and consider a sequence $(\phi_n)$ in $\mathcal{C}_c^\infty({\ensuremath{\mathcal O}})$, $0\leq\phi_n\leq1$, converging to 1 everywhere on ${\ensuremath{\mathcal O}}$. By Fatou’s lemma, $$0\leq\int_{[0,T[\times{\ensuremath{\mathcal O}}}{{1} \hspace{-0.25 em}{\rm I}}_A
d\nu(x,t)\leq\liminf_{n\rightarrow\infty}\int_{[0,T[\times{\ensuremath{\mathcal O}}}{{1} \hspace{-0.25 em}{\rm I}}_A\phi_nd\nu(x,t)=0.$$

We end this part by a convergence lemma which plays an important role in our approach (Lemma 3.8 in [@PIERRE]):

\[convergemeas\] If $v^n\in\mathcal{P}$ is a bounded sequence in $\mathcal{K}$ and converges weakly to $v$ in $L^2([0,T];H_0^1(\mathcal{O}))$; if $u$ is a quasi-continuous function and $|u|$ is bounded by a element in $\mathcal{P}$. Then $$\lim_{n\rightarrow+\infty}\int_0^T\int_\mathcal{O}ud\nu^{v^n}=\int_0^T\int_\mathcal{O}ud\nu^{v}.$$

We now give the assumptions on the obstacle that we shall need in the different cases that we shall consider.\
**Assumption (O):** The obstacle $S: [0,T]\times \Omega\times
{\ensuremath{\mathcal O}}\rightarrow {\mathbb{R}}$ is an adapted random field almost surely quasi-continuous, in the sense that for $P$-almost all 