The Unapologetic Mathematician

Mathematics for the interested outsider

The Radon-Nikodym Theorem for Signed Measures

Now that we’ve proven the Radon-Nikodym theorem, we can extend it to the case where \mu is a \sigma-finite signed measures.

Indeed, let X=A\uplus B be a Hahn decomposition for \mu. We find that \mu^+ is a \sigma-finite measure on A, while \mu^- is a \sigma-finite measure on B.

As it turns out that \nu\ll\mu^+ on A, while \nu\ll\mu^- on B. For the first case, let E\subseteq A be a set for which \mu^+(E)=0. Since E\cap B=\emptyset, we must have \mu^-(E)=0, and so \lvert\mu\rvert(E)=\mu^+(E)+\mu^-(E)=0. Then by absolute continuity, we conclude that \nu(E)=0, and thus \nu\ll\mu^+ on A. The proof that \nu\ll\mu^- on B is similar.

So now we can use the Radon-Nikodym theorem to show that there must be functions f_A on A and f_B on B so that

\displaystyle\begin{aligned}\nu(E\cap A)=&\int\limits_{E\cap A}f_A\,d\mu^+\\\nu(E\cap B)=&\int\limits_{E\cap B}f_B\,d\mu^-=-\int\limits_{E\cap B}-f_B\,d\mu^-\end{aligned}

We define a function f on all of X by f(x)=f^+(x) for x\in A and f(x)=f^-(x) for x\in B. Then we can calculate

\displaystyle\begin{aligned}\nu(E)&=\nu((E\cap A)\uplus(E\cap B))\\&=\nu(E\cap A)+\nu(E\cap B)\\&=\int\limits_{E\cap A}f_A\,d\mu^+-\int\limits_{E\cap B}-f_B\,d\mu^-\\&=\int\limits_{E\cap A}f\,d\mu^+-\int\limits_{E\cap B}f\,d\mu^-\\&=\int\limits_Ef\,d\mu\end{aligned}

which in exactly the conclusion of the Radon-Nikodym theorem for the signed measure \mu.

About these ads

July 8, 2010 - Posted by | Analysis, Measure Theory

5 Comments »

  1. [...] Radon-Nikodym Derivative Okay, so the Radon-Nikodym theorem and its analogue for signed measures tell us that if we have two -finite signed measures and with , then there’s some function [...]

    Pingback by The Radon-Nikodym Derivative « The Unapologetic Mathematician | July 9, 2010 | Reply

  2. here, there are several formulas which does not display right

    Comment by juanmarqz | July 13, 2010 | Reply

  3. They’re all displaying correctly for me, at least for now. WordPress’ \LaTeX support has been extremely buggy the last week or so. Is it saying something like “latex path not specified”?

    Comment by John Armstrong | July 13, 2010 | Reply

  4. yes “latex path not specified” in red and yellow background…

    Comment by juanmarqz | July 13, 2010 | Reply

  5. They tend to be transitory. Try force-reloading every so often.

    If nothing else, mouseover should show the \LaTeX source.

    Comment by John Armstrong | July 13, 2010 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 392 other followers

%d bloggers like this: