Let $W$ be an isonormal process over a real separable Hilbert space. Let $D, \delta$ be the associated Malliavin derivative, divergence operators.
Exercise 1.3.4.
Let $F$ be in $\mathbb{D}^{1,2}$ (meaning $F\in L^2(\Omega)$ and $\mathbb{E}\big[ || DF||_H \big] < \infty$)
suppose $\mathbb{E}[ |F|^{-2} ]$ is finite.
Show $\mathbb{P}(F>0)\in\{0,1\}$.
It is clear that $\mathbb{P}(F =0 ) = 0$. And the hint suggests
(i) we take $\psi_\epsilon$ an approximation of the sign function $$ {\rm sign}(x) = 1_{\{x>0 \}} - 1_{\{x < 0 \}}. $$ and
(ii) take $u$ (arbitrary) in $\text{domain}(\delta)$ with $\| u \|_H$ uniformly bounded
and
(iii) use duality relation to compute
$$ \mathbb{E}\big[ \psi_\epsilon(F) \delta(u) \big] = \mathbb{E}\big[ \langle u, DF \rangle_H \psi_\epsilon'(F) \big.] $$ Left side converges to $\mathbb{E}[ {\rm sign}(F) \delta(u) ]$.
(A) Right side seems to converge to zero
so that we would have $$ \mathbb{E}[ {\rm sign}(F) \delta(u) ] =0. \quad(\#) $$ The above equality holds for arbitrary $u$ with a.s. bounded $\| u\|_H$.
(B) This (#) seems to imply ${\rm sign}(F)$ is constant, which implies $\mathbb{P}(F>0)\in\{0,1\}$
For (A): $\psi_\epsilon'(F)\to 0$ almost surely since $|F|>0$ a.s. But I do not know how to justify the convergence of the expectation, ($\psi_\epsilon'(F)$ is not uniformly bounded as $\epsilon\downarrow 0$.) I do not know how to use the assumption $\mathbb{E} |F|^{-2}<\infty$.
For (B): So the set of random variables of the form $\delta(u)$, with $\| u\|_H$ a.s. bounded, is $L^2$-orthogonal complement of the space of constant random variables?
Any comment or clarification is appreciated. Thanks.