On Conditional Jensen Inequality Hypothesis

Question

** Conditional Jensen Inequality

Let $X$ be in $L^1$, and $\mathcal{G}$ a sigma-algebra of the space and $\phi$ a convex function. Then, $\mathbb E(\phi(X)|\mathcal{G}) \geq \phi(\mathbb E(X|\mathcal{G}))$.

**

Instead of requiring $X$ to be in $L^1$, can we weaken the hypothesis and just require $\mathbb E(\phi(X)|\mathcal{G})< \infty$?

Answer 1

If $X$ is not in $L^1$, the RHS of the inequality is not defined. Suppose for example that $X$ is Cauchy distributed and that $\sigma(X)$ and $\mathcal{G}$ are independent.