If $A\subseteq\Omega$, where $\mathbb{P}^{*}(A)=0$, then $A\in\mathcal{M}$. For $([0,1],\mathcal{M},\lambda)$, a set $H\in\mathcal{M}$ has measure zero iff $$\inf\left\{ \sum_{i}\lambda(A_{i}):A_{i} \text{ intervals},\cup_{i}A_{i}\supseteq H \right\}=0$$