Not all sets in $\mathbb{R}$ are Lebesgue measurable sets

Question

Let $\mathcal{B}(\mathbb{R})$ be the Borel-sigma algebra on $\mathbb{R}$, and let $\mathcal{L}$ be the Lebesgue sigma-algebra which is a unique extension of $\mathcal{B}(\mathbb{R})$ with respect to the length measure $m$.

I'm trying to find a counter-example for each for the following:

1. $\mathcal{L}\neq \mathbb{P}(\mathbb{R})$
2. $\mathcal{L}\neq \mathcal{B}(\mathbb{R})$

For $2$, I just need to find a null set (has length $0$) that is not a member of $\mathcal{B}(\mathbb{R})$. I thought about the Cantor set because it has measure zero, but it's apparently Borel measurable.