## The Radon-Nikodym Theorem for Signed Measures

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

Indeed, let be a Hahn decomposition for . We find that is a -finite measure on , while is a -finite measure on .

As it turns out that on , while on . For the first case, let be a set for which . Since , we must have , and so . Then by absolute continuity, we conclude that , and thus on . The proof that on is similar.

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

We define a function on all of by for and for . Then we can calculate

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

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[…] 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 |

here, there are several formulas which does not display right

Comment by juanmarqz | July 13, 2010 |

They’re all displaying correctly for me, at least for now. WordPress’ 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 |

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

Comment by juanmarqz | July 13, 2010 |

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

If nothing else, mouseover should show the source.

Comment by John Armstrong | July 13, 2010 |