Expectation is Monotone

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

Let $X$ and $Y$ be integrable random variables such that:


 * $\forall \omega \in \Omega: \map X \omega \le \map Y \omega$

Then:


 * $\expect X \le \expect Y$

Proof
From the definition of expectation we have:


 * $\ds \expect X = \int X \rd \Pr$

and:


 * $\ds \expect Y = \int Y \rd \Pr$

The result follows directly from Integral of Integrable Function is Monotone.