Theorem

Let $(X,\mathcal{F},\mu)$ be a measure space. Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of measurable functions, $f_{n}:X\to \mathbb{R}$. Let $f:X\to \mathbb{R}$ be such that $f_{n}\to f$ pointwise, then $$f:X\to \mathbb{R}\text{ is measurable}$$