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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?

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up vote 6 down vote accepted

The concave function $f$ is an pointwise infimum of a countable family $(a_n)$ of affine functions. You can arrange for the sequence $(f_n)$ to contain each $a_n\circ X$ infinitely often, so that $\liminf_{n\to\infty} f_n(\omega)=\inf_n a_n(X(\omega))$. Now the left hand sides of the two inequalities are the same, and so are the right hand sides, since the affinity of $a_n$ implies $E(f_n)=E(a_n\circ X)=a_n(E(X))$.

However, Fatou's lemma contains much more information, as there is really no limit operation in this construction, just the infimum. So the connection is pretty much one way.

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