Measure of $\{t:B_t\in E\}$ for some null set $E$.

Question

I am wondering if the following result can be found in any textbook or if you have a proof of it.

When $E$ is a null set and $B_t$ is the Brownian motion, we have almost surely : $$\mathcal{L}^1\{t:B_t\in E\}=0.$$

Here, I denoted by $\mathcal{L}^1$ the $1$-dimensional Lebesgue measure. This result is well known when $E=\{x\}$ is any singleton.

Answer 1

Since the distribution of $B_t$ is absolutely continuous (with respect to Lebesgue measure), if $E$ is has measure zero (with respect to Lebesgue measure) then $\mathbb P(B_t\in E)=0$. By Fubini, $$ \mathbb E(\mathcal L^1\{t:B_t\in E\})=\displaystyle\int_0^{+\infty}\mathbb P(B_t\in E)\,\mathrm dt=0. $$ Since $\mathcal L^1\{t:B_t\in E\}\geqslant0$ almost surely, this implies that $\mathcal L^1\{t:B_t\in E\}=0$ almost surely.