Let $E\subset B_1(0)\subset \mathbb{R}^n$ be a compact set s.t. $\lambda(E)=0$, where $\lambda$ is the Lebesgue measure, and $B_1(0)$ is the Euclidean unit ball centered at the origin. Is the following integral finite:

$$\int_{B_1(0)}-\log d(x,E)d\lambda(x)<\infty?$$

Although this question seems trivial, I have failed to find a reference to it or to variations of it in previous discussions. I was not able to come up with a counter-example nor a proof. I also asked in mathstackexchange a variation of it, but didn’t get a sufficient answer.

Thanks ahead