If the measure $\lambda(E)$ of a set $E$ is finite then $E$ is measurable iff for each $\varepsilon>0$, there exist disjoint finite intervals $I_1,I_2,\dots,I_n$ such that $\lambda(E \triangle \bigcup_{k=1}^n I_k) \leq \varepsilon$ (where $\triangle$ denotes the symmetric difference, and $\lambda$ is the Lebesgue measure).
Can anyone provide me with a counterexample for the case when measure of $E$ is infinite?
Take $E=\mathbb{R}$. Then, any finite collection of finite intervals will have measure at most n times the largest one, so the symmetric difference will contain the interval $(-\infty, c-1]$ where $c = \inf_{x \in \cup_{i=1}^n I_n} x$.