I am solving Exercise 8.4 from Bernt Øksendal's "Stochastic Differential Equations":
Let $B_t$ be $n$-dimensional Brownian motion and let $F\subset \mathbb R^n$ be a Borel set. Prove that the expected total length of time that $B_t$ spends in $F$ is zero if and only if $F$ has zero $n$-dimensional Lebesgue measure.
The hint is to consider the Resolvent $R_\alpha$ for $\alpha>0$ and let $\alpha \to 0$.
I define the expected total length of time $t$ that $B_t$ stays in $F$ by $$ E^0\bigg[ \int\limits_0^\infty 1_F(B_t) dt\bigg]. \tag{1} $$ Hence Fubini's theorem implies the "if" part. But I am not sure how to rigorously argue the the only if part.
My idea: Suppose (1) is 0. Then for any give $x$, by shifting the set $F$ and the Brownian motion we obtain that $$R_\alpha 1_F(x)=E^x\bigg[ \int\limits_0^\infty e^{-\alpha t}1_F(B_t) dt\bigg]=0, \quad \forall x\in \mathbb{R}^n.$$ Then if I could modify $1_F$ by some $g\in C_b(\mathbb{R}^n)$, then basically by left-multiplying $(\alpha-A)$, and using the identity $$(\alpha-A)R_\alpha g=g,\quad g\in C_b,$$ I can show $1_F\equiv 0$.
I am not sure whether my understanding is correct and how to convert it to a rigorous argument. Obviously, I haven't used the hint.