in $L^p({\mathbb{R}})$, where these notions are defined as follows:

-   For any $h$ in $H$, there exists a r.v. $\tilde{F}_h$ such that $\tilde{F}_h=F$, $\P_0-$a.s., and such that for any $\omega$ in $\Omega$, $t \in {\mathbb{R}}\longmapsto \tilde{F}_h(\omega+t h)$ is absolutely continuous, where $t h:=(t h^1,\ldots,t h^d)$.

-   There exists $\mathcal{D}F$ in $L^p(H)$ such that for any $h$ in $H$, $$\label{eq:cvproba}
    \frac{F{\circ \tau_{\varepsilon h}}-F}{\varepsilon} \underset{\varepsilon \to 0}{\longrightarrow} \langle \mathcal{D}F, h\rangle_H,\ \text{in probability.}$$

In addition, for any $F$ in $\D^{1,p}$, $\nabla F=\mathcal{D}F$, $\P_0-$a.s. Note that according to the statement of Step 1 in the proof of [@Sugita Theorem 3.1], if $F$ is (RAC) and (SGD) then for any $h$ in $H$ and any $\varepsilon>0$ it holds that $$\varepsilon^{-1}(\tilde{F}_h{\circ \tau_{\varepsilon h}} -\tilde{F}_h) =\varepsilon^{-1} \int_0^\varepsilon \langle \nabla F {\circ \tau_{s h}}, h\rangle_H ds, \ \P_0-a.s.$$ Furthermore, by Lemma \[lemma:UstunelZakai\], we have for any $\varepsilon$ there exists a set $A^\varepsilon$ such that $\P_0[A^\varepsilon]=0$ and $F{\circ \tau_{\varepsilon h}}=\tilde{F}_h {\circ \tau_{\varepsilon h}}$ and $F=\tilde F_h$ outside $A^\varepsilon$. Hence, for any $\varepsilon$ in $(0,1)$, the relation above rewrites as: $$\label{eq:cons}
\varepsilon^{-1} (F{\circ \tau_{\varepsilon h}}-F) = \varepsilon^{-1} \int_0^\varepsilon \langle \nabla F {\circ \tau_{s h}}, h\rangle_H ds, \ \P_0-a.s.$$

It has actually been proved by Janson [@janson] that is equivalent to 