Conditional Expectation is Non-Decreasing

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a Probability Space and let $\mathcal{B} \subset \mathcal{G}$ be a sub-Sigma Algebra of $\mathcal{G}$. Suppose $X_{1}, X_{2} \in \mathcal{G}$ so that $X_{1} \leq X_{2}$. Then Almost Surely $$\mathbb{E}(X_{1} | \mathcal{B}) \leq \mathbb{E}(X_{2} | \mathcal{B})$$

Recall that A Measureable Function is less than or equal to another iff all its integrals are less than or equal to the others. Therefore, $\forall A \in \mathcal{B}$, by definition of Conditional Expectation with respect to Sigma Field: $$\int\limits_{A} \mathbb{E}(X_{1} | \mathcal{B}) d \mathbb{P} = \int\limits_{A} X_{1} d \mathbb{P} \leq \int\limits_{A} X_{2} d \mathbb{P} = \int\limits_{A} \mathbb{E}(X_{2} | \mathcal{B}).$$ Then, again because A Measureable Function is less than or equal to another iff all its integrals are less than or equal to the others, we have that $$\mathbb{E}(X_{1} | \mathcal{B}) \leq \mathbb{E}(X_{2} | \mathcal{B})$$ Almost Surely. $\blacksquare$