The Unapologetic Mathematician

Mathematics for the interested outsider

The Radon-Nikodym Derivative

Okay, so the Radon-Nikodym theorem and its analogue for signed measures tell us that if we have two \sigma-finite signed measures \mu and \nu with \nu\ll\mu, then there’s some function f so that

\displaystyle\nu(E)=\int\limits_Ef\,d\mu

But we also know that by definition

\displaystyle\nu(E)=\int\limits_E\,d\nu

If both of these integrals were taken with respect to the same measure, we would know that the equality

\displaystyle\int\limits_Ef\,d\mu=\int\limits_Eg\,d\mu

for all measurable E implies that f=g \mu-almost everywhere. The same thing can’t quite be said here, but it motivates us to say that in some sense we have equality of “differential measures” d\nu=f\,d\mu. In and of itself this doesn’t really make sense, but we define the symbol

\displaystyle\frac{d\nu}{d\mu}=f

and call it the “Radon-Nikodym derivative” of \nu by \mu. Now we can write

\displaystyle\int\limits_E\,d\nu=\int\limits_Ef\,d\mu=\int\limits_E\frac{d\nu}{d\mu}\,d\mu

The left equality is the Radon-Nikodym theorem, and the right equality is just the substitution of the new symbol for f. Of course, this function — and the symbol \frac{d\nu}{d\mu} — is only defined uniquely \mu-almost everywhere.

The notation and name is obviously suggestive of differentiation, and indeed the usual laws of derivatives hold. We’ll start today by the easy property of linearity.

That is, if \nu_1 and \nu_2 are both \sigma-finite signed measures, and if a_1 and a_2, then a_1\nu_1+a_2\nu_2 is clearly another \sigma-finite signed measure. Further, it’s not hard to see if \nu_i\ll\mu then a_1\nu_1+a_2\nu_2\ll\mu as well. By the Radon-Nikodym theorem we have functions f_1 and f_2 so that

\displaystyle\begin{aligned}\nu_1(E)&=\int\limits_Ef_1\,d\mu\\\nu_2(E)&=\int\limits_Ef_2\,d\mu\end{aligned}

for all measurable sets E. Then it’s clear that

\displaystyle\begin{aligned}\left[a_1\nu_1+a_2\nu_2\right](E)&=a_1\nu_1(E)+a_2\nu_2(E)\\&=a_1\int\limits_Ef_1\,d\mu+a_2\int\limits_Ef_2\,d\mu\\&=\int\limits_Ea_1f_1+a_2f_2\,d\mu\end{aligned}

That is, a_1f_1+a_2f_2 can serve as the Radon-Nikodym derivative of a_1\nu_1+a_2\nu_2 with respect to \mu. We can also write this in our suggestive notation as

\displaystyle\frac{d(a_1\nu_1+a_2\nu_2)}{d\mu}=a_1\frac{d\nu_1}{d\mu}+a_2\frac{d\nu_2}{d\mu}

which equation holds \mu-almost everywhere.

July 9, 2010 Posted by | Analysis, Measure Theory | 3 Comments

   

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