Let $M = (X, \mathcal{F}, \mu)$ be a
D1158: Measure space such that
| (i) |
$f_0, f_1, f_2, \cdots : X \to [- \infty, \infty]$ are each a D5600: Basic Borel function on $M$
|
| (ii) |
\begin{equation}
\forall \, n \in \mathbb{N} :
f_n \overset{a.e.}{\geq} 0
\end{equation}
|
Then
| (1) |
\begin{equation}
\int_X \liminf_{n \to \infty} f_n \,d \mu \leq \liminf_{n \to \infty} \int_X f_n \,d \mu
\end{equation}
|
| (2) |
\begin{equation}
\int_X \limsup_{n \to \infty} f_n \,d \mu \geq \limsup_{n \to \infty} \int_X f_n \,d \mu
\end{equation}
|
