User:Caliburn/s/prob/Expectation is Monotone

Theorem

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

Let $X$ and $Y$ be integrable random variables on $\struct {\Omega, \Sigma, \Pr}$ such that:

$\map X \omega \le \map Y \omega$

for each $\omega \in \Omega$.

Then:

$\expect X \le \expect Y$

Proof

From Pointwise Difference of Measurable Functions is Measurable, we have:

$Y - X$ is $\Sigma$-measurable.

From Integral of Integrable Function is Additive: Corollary 2, we have:

$Y - X$ is integrable.

We also have:

$\map {\paren {Y - X} } \omega \ge 0$ for each $\omega \in \Omega$.

So, from Expectation of Non-Negative Random Variable is Non-Negative: Corollary, we have:

$\expect {Y - X} \ge 0$

From Expectation is Linear we then obtain:

$\expect Y - \expect X \ge 0$

that is:

$\expect X \le \expect Y$