Expectation of Almost Surely Constant Random Variable

Theorem

Let $X$ be an almost surely constant random variable.

That is, there exists some $c \in \R$ such that:

$\map \Pr {X = c} = 1$

Then:

$\expect X = c$

Proof

Note that since $\map \Pr {X = c} = 1$, we have $\map \Pr {X \ne c} = 0$ from Probability of Event not Occurring.

Therefore:

$\map {\mathrm {supp} } X = \set c$

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We therefore have:

$\blacksquare$