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I'm often reading this version of the Jensen inequality for conditional expectation:

Let $(\Omega,\mathcal A, P)$ a probability space and $X$ a integrable random variable. Then for any convex function $\phi:\mathbb R\to\mathbb R$ holds $$\phi\big(E(X|\mathcal A)\big)\leq E\big(\phi(X)|\mathcal A\big).$$

I'm wondering if there isn't the requirement "$\phi(X)$ is integrable" missing? Otherwise I can't imagine why the RHS should even exist.

For the ordinary case (not conditioned) I understand why we don't need $\phi(X)$ to be integrable, as the RHS would be $\infty$ and the inequality would hold trivially. But here we might have something on the RHS that doesn't even exist.

mag
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1 Answers1

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Because $\phi$ is convex, it is bounded below by a function of the form $x\mapsto mx+b$, so the integrability of $X$ implies that of $\phi(X)^-$ (the negative part of $\phi(X)$). Even though $\phi(X)^+$ may not be integrable, the conditional expectation $E[\phi(X)\mid\mathcal A]$, as a random variable with values in $(-\infty,+\infty]$, exists in a generalized well-defined sense (for example, as the increasing limit $\lim_nE[\phi(X)\wedge n\mid\mathcal A]$). As such Jensen's inequality holds, even on the event $\{E[\phi(X)\mid\mathcal A]=+\infty\}$ because $\phi(E[X\mid\mathcal A])$ is a.s. finite.

John Dawkins
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