MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.

share|cite|improve this question

closed as off topic by Andrés E. Caicedo, Michael Renardy, Anthony Quas, Alexandre Eremenko, Goldstern Dec 10 '12 at 23:58

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Have a look at the FAQ. This question does not belong here. – Anthony Quas Dec 10 '12 at 23:49
This is a wrong place for this sort of questions. Please, read FAQ. – Alexandre Eremenko Dec 10 '12 at 23:52
up vote 2 down vote accepted

Let e.g. $\nu$ be a finite Borel measure on $[a,b]$ and $m$ the Lebesgue measure. So the function $[a,b]\ni x\mapsto \nu\big([a,x)\big)$ is a BV function. It is therefore differentiable $m$-a.e., and its derivative $\rho(x)$ coincides with the Radon-Nikodym derivative of the absolutely continuous part of $\nu$. Knowing $\rho$ you can deduce the Lebesgue decomposition of $\nu$ wrto the Lebesgue measure: $\nu=\nu _ a+\nu _ s$ with $\nu _a\perp\nu _ s$ where $\nu_a(E)=\int_E\rho(x)dm(x)$.

share|cite|improve this answer
"Pietro Majer", thsnk you a lot. – Omid Saba Dec 11 '12 at 4:34

Not the answer you're looking for? Browse other questions tagged or ask your own question.