Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable

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

Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $P_X$ be the probability distribution of $X$.

Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.

Let $h : \R \to \R$ be a $\map \BB \R$-measurable function.

Let $f_X$ be a probability density function for $X$.

Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.

Then $\map h X$ is integrable :


 * $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$

and in this case:


 * $\ds \expect {\map h X} = \int_\R \map h x \map {f_X} x \rd \map \lambda x$

Proof
From Composition of Measurable Mappings is Measurable:


 * $\map h X$ is $\Sigma$-measurable.

So:


 * $\map h X$ is a real-valued random variable.

From Characterization of Integrable Functions, we have that:


 * $\map h X$ is integrable $\size {\map h X}$ is integrable.

We have:

Then:


 * $\ds \int \size {\map h X} \rd \Pr$ $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$

so:


 * $\size {\map h X}$ is integrable $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$.

So:


 * $\map h X$ is integrable $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$.

If $\map h X$ is integrable, then we have: