 Change of measure

A change of measure replaces a positive measure $\mu$ by a positive measure $\nu$ described by a nonnegative Radon-Nikodym derivative $w=d\nu /d\mu$. For a nonnegative measurable function $g$, or for a function $g$ that is integrable with respect to $\nu$,Thus the integration weight changes while the underlying measurable space stays fixed. A density representation requires absolute continuity of measures $\nu \ll \mu$; the Radon-Nikodym theorem provides such a density under its hypotheses.

On $X=\{0,1\}$, let $\mu$ assign mass $1/2$ to each point and let $\nu$ assign masses $1/4,3/4$. Then $w(0)=1/2$ and $w(1)=3/2$, soFor probability measures, the identity becomes ${\mathbb{E}}_{\nu }[g]={\mathbb{E}}_{\mu }[wg]$ and ${\mathbb{E}}_{\mu }[w]=1$. It underlies likelihood ratios and importance sampling; strict positivity $\mu$-almost everywhere makes the two probability measures equivalent.