Theorem (1.28)

Let $(X,\mathcal{F},\mu)$ be a measure space. Let Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of measurable functions, $f_{n}:X\to \mathbb{R}^{+}$. Then, $$\int\limits_{X} \liminf_{ n \to \infty } f_{n} \, d\mu\le\liminf_{ n \to \infty } \int\limits_{X} f_{n} \, d\mu$$