Does a converging sequence of measurable functions necessarily converge towards a measurable function?

Question

Say $f_n\geq0\ \forall n$ are measurable in the sense that $\{x\ :\ f(x)<\alpha\}\in\mathcal M\ \forall\alpha\in\Bbb R$ where $\mathcal M$ is the set of Lebesgue measurable subsets of $\Bbb R$. And $f_n\rightarrow f$ Is $f :\Bbb R\mapsto\Bbb R$ necessarily a measurable function in the same sense?