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Radon-Nikodym theorem

Let $(X,ℬ,\mu )$ be a sigma-finite? measure space and let $\nu$ be a measure defined on $ℬ$ which is absolutely continuous with respect to $\mu$. Then there is a nonnegative measurable function $f$ such that for each set $E$ in $ℬ$ we have

Furthermore, $f$ is unique a.e.? $\mu$. The function $f$ is referred to as the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ and is sometimes denoted by $\left[\frac{d\nu }{d\mu }\right]$.

References

• Royden, H.L. (1988): Real Analysis. Prentice Hall.