For $f_n\in L^+$, the integral of $\liminf f_n$ is at most the $\liminf$ of the integrals.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space and let \(f_n\) for \(n\geq1\) belong to $L^+$. Then \[ \int_X \liminf_{n\to\infty} f_n \dmu \leq \liminf_{n\to\infty} \int_X f_n \dmu . \]
The inequality can be strict. On \(\RR\) with Lebesgue measure, let \(f_n\coloneqq\one_{[n,n+1]}\). Then \(\liminf_n f_n=0\) pointwise, so the left side is \(0\), while \(\int_\RR f_n=1\) for every \(n\).
State Fatou's lemma.