Monotone Convergence Theorem

Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~Measure Space) and let $0 \leq f_1 \leq f_2 \leq \cdots$ be a non-decreasing sequence of non-negative measurable functions with $f_n \to f$ pointwise. Then \[ \lim_{n \to \infty} \int f_n\, d\mu = \int f\, d\mu. \]