Conditional Jensen's Inequality

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.

Let $X$ be an integrable random variable.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Let $f : \R \to \R$ be a convex function such that $\map f X$ is integrable.

Let $\expect {\map f X \mid \GG}$ be a version of the conditional expectation of $\map f X$ given $\GG$.

Then:


 * $\map f {\expect {X \mid \GG} } \le \expect {\map f X \mid \GG}$

Proof
Note that from Convex Real Function is Measurable and Composition of Measurable Mappings is Measurable, $\map f X$ is $\Sigma$-measurable, so the hypotheses make sense.

By Convex Real Function is Pointwise Supremum of Affine Functions: Corollary, there exists a countable set $\SS \subseteq \R^2$ such that:


 * $\ds \map f x = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a x + b}$

so:


 * $\ds \map f X = \sup_{\tuple {a, b} \mathop \in \SS} \paren {a X + b}$

So for each $\tuple {a, b} \in \SS$, we have:


 * $a X + b \le \map f X$

From Conditional Expectation is Monotone and Conditional Expectation is Linear, we have:


 * $a \expect {X \mid \GG} + b \le \expect {\map f X \mid \GG}$ almost surely

for each $\tuple {a, b} \in \SS$.

For each $\tuple {a, b} \in \SS$, let $A_{\tuple {a, b} }$ be the set of sample points $\omega \in \Omega$ such that this inequality fails.

Then for:


 * $\ds \omega \in \Omega \setminus \bigcup_{\tuple {a, b} \mathop \in \SS} A_{\tuple {a, b} }$

we have:


 * $a \map {\expect {X \mid \GG} } \omega + b \le \map {\expect {\map f X \mid \GG} } \omega$ for all $\tuple {a, b} \in \SS$

From Null Sets Closed under Countable Union, we have:


 * $\ds \map \Pr {\bigcup_{\tuple {a, b} \mathop \in \SS} A_{\tuple {a, b} } } = 0$

So we have:


 * $a \expect {X \mid \GG} + b \le \expect {\map f X \mid \GG}$ for all $\tuple {a, b} \in \SS$ almost surely.

We therefore have:


 * $\ds \sup_{\tuple {a, b} \mathop \in \SS} \paren {a \expect {X \mid \GG} + b} \le \expect {\map f X \mid \GG}$ almost surely.

that is:


 * $\map f {\expect {X \mid \GG} } \le \expect {\map f X \mid \GG}$ almost surely.