Expectation is Monotone

Theorem

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

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

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

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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.

$\blacksquare$