Category:Change of Measures Formula for Integrals

This category contains pages concerning Change of Measures Formula for Integrals:

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 positive $\Sigma$-measurable function.

Then:

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

where:

$f \cdot g$ is the pointwise product of $f$ and $g$

$\ds \int \cdot \rd \nu$ denotes the integral of a positive $\Sigma$-measurable function with respect to $\nu$.