O\supset E\ open\}$.

A property is said to hold quasi-everywhere (in short q.e.) if it holds outside a set of null capacity.

(Quasi-continuous)

A function $u:[0,T[\times\mathcal{O}\rightarrow\mathbb{R}$ is called quasi-continuous, if there exists a decreasing sequence of open subsets $O_n$ of $[0,T[\times\mathcal{O}$ with:

1.  for all $n$, the restriction of $u_n$ to the complement of $O_n$ is continuous;

2.  $\lim_{n\rightarrow+\infty}cap\;(O_n)=0$.

We say that $u$ admits a quasi-continuous version, if there exists $\tilde{u}$ quasi-continuous such that $\tilde{u}=u\ a.e.$

The next proposition, whose proof may be found in [@Pierre] or [@PIERRE] shall play an important role in the sequel:

\[Versiont\] Let $K\subset {\ensuremath{\mathcal O}}$ a compact set, then $\forall t\in [0,T[$ $$cap (\{ t\}\times K)=\lambda_d (K),$$ where $\lambda_d$ is the Lebesgue measure on ${\ensuremath{\mathcal O}}$.\
As a consequence, if $u: [0,T[\times {\ensuremath{\mathcal O}}\rightarrow {\mathbb{R}}$ is a map defined quasi-everywhere then it defines uniquely a map from $[0,T[$ into $L^2 ({\ensuremath{\mathcal O}})$. In other words, for any $t\in [0,T[$, $u_t$ is defined without any ambiguity as an element in $L^2 ({\ensuremath{\mathcal O}})$. Moreover, if $u\in \mathcal{P}$, it admits version $\bar{u}$ which is left continuous on $[0,T]$ with values in $L^2 ({\ensuremath{\mathcal O}})$ so that $u_T =\bar{u}_{T^-}$ is also defined without ambiguity.

The previous proposition 