> [!theorem] > > Let $(\Omega, \cf, \bp)$ be a [[Probability|probability space]] and $X$ be a [[Random Variable|random variable]]. Then for all [[Borel Measurable Function|Borel measurable]] functions $g: \real \to \real$ with $g(X) \in L^1$ [[Integrable Function|integrable]], > $ > \ev(g(X)) = \int g(x)\mu_X(dx) > $ > where $\mu_X$ is the [[Probability Distribution|probability distribution]] of $X$. > > Moreover, if $X$ has [[Dense|density]]/[[Lebesgue-Radon-Nikodym Theorem|Radon-Nikodym derivative]] $f$ with respect to the [[Lebesgue Measure|Lebesgue measure]], then > $ > \ev(g(X)) = \int g(x)f(x)dx = \int g \frac{d\mu_X}{dm}dm > $ > *Proof*. First note that the above holds for all indicator functions by definition of the integral. By linearity and [[Monotone Convergence Theorem|MCT]], we can apply a [[Monotone Class Argument]] to show that the above holds for all Borel-measurable functions.