Change of Measures Formula for Integrals/Corollary

Corollary
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$ such that:


 * $\nu$ is absolutely continuous with respect to $\mu$.

Let $g$ be a Radon-Nikodym derivative of $\nu$ with respect to $\mu$. Let $f : X \to \overline \R$ be a $\nu$-integrable function.

Then $f \cdot g$ is $\mu$-integrable with:


 * $\ds \int f \rd \nu = \int \paren {f \cdot g} \rd \mu$

where:


 * $f \cdot g$ is the pointwise product of $f$ and $g$
 * $\ds \int \cdot \rd \nu$ denotes the integral of a $\nu$-integrable function with respect to $\nu$.

Proof
From Pointwise Product of Measurable Functions is Measurable, we have:


 * $f \cdot g$ is $\Sigma$-measurable.

We show that:


 * $\ds \int \paren {f \cdot g}^+ \rd \mu < \infty$

and:
 * $\ds \int \paren {f \cdot g}^- \rd \mu < \infty$

where $\paren {f \cdot g}^+$ and $\paren {f \cdot g}^-$ denote the positive part and negative part of $f \cdot g$ respectively.

We have:

and:

So $f \cdot g$ is $\mu$-integrable.

Then: