I would like to know if there is any relationship between Fatou's lemma and Jensen's inequality. On paper, I find a similarity in their expressions.
- Fatou's Lemma: $\int (\liminf_{n \to \infty}f_n)\le \liminf_{n \to \infty}\int (f_n) $
- Jensen's inequality: $\mathbb{E}(f(X)) \le f(\mathbb{E} (X))$ if $f(x)$ is concave in $x$.
Could someone provide me with some insight explaining this similarity?