Intuition behind the expectation definition

Question

What is the intuition behind the general definition of expectation for a measurable random variable X ? Please help.

Answer 1

The intuition is that integrals with respect to measures are the mathematical generalization of sums.

Consider the example of $X$ being the value of a die roll. Then we know that

$$ E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = 3.5$$

This is a sum, as we were taught in beginner probability that expectations are sums. But in measure theory we learn that sums are just a type of integral. Suppose we view $X$ as being a measurable function on a discrete measure space $\Omega = \{1,2,3,4,5,6\}$, $\Sigma$ being the power set of $\Omega$, and with discrete measure $P$ defined by $P(1) = \ldots = P(6) = 1/6$. Then $E(X)$ can also be written as

$$ E(X) = \int_{\Omega} X \, dP $$

as given in your generalized definition. This allows us to generalize expectations to random variables on a continuous domain.