The Unapologetic Mathematician

Mathematics for the interested outsider

Corollaries of the Chain Rule

Today we’ll look at a couple corollaries of the Radon-Nikodym chain rule.

First up, we have an analogue of the change of variables formula, which was closely tied to the chain rule in the first place. If \lambda and \mu are totally \sigma-finite signed measures with \mu\ll\lambda, and if f is a finite-valued \mu-integrable function, then

\displaystyle\int f\,d\mu=\int f\frac{d\mu}{d\lambda}\,d\lambda

which further justifies the the substitution of one “differential measure” for another.

So, define a signed measure \nu as the indefinite integral of f. Immediately we know that \nu is totally \sigma-finite and that \nu\ll\mu. And, obviously, f is the Radon-Nikodym derivative of \nu with respect to \mu. Thus we can invoke the above chain rule to conclude that \lambda-a.e. we have

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

We then know that for every measurable E

\displaystyle\nu(E)=\int\limits_Ef\frac{d\mu}{d\lambda}\,d\lambda

and the substitution formula follows by putting X in for E.

Secondly, if \mu and \nu are totally \sigma-finite signed measures so that \mu\equiv\nu — that is, \mu\ll\nu and \nu\ll\mu — then \mu-a.e. we have

\displaystyle\frac{d\mu}{d\nu}\frac{d\nu}{d\mu}=1

Indeed, \mu\ll\mu, and by definition we have

\displaystyle\mu(E)=\int\limits_E1\,d\mu

so 1 serves as the Radon-Nikodym derivative of \mu with respect to itself. Putting this into the chain rule immediately gives us the desired result.

July 13, 2010 Posted by | Analysis, Measure Theory | 6 Comments

   

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