Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is a question that I came across while learning measure theory. As a matter of fact I am completely new to this area so I do not really know how to approach it. Some hints or perhaps guidelines to solve this questions will be very appreciated.

Let $f\in L^+(X,M)$ and let $\mu$ be a positive measure on $M$. For $E\in M$, define $\lambda(E)=\int _{E}f d\mu$. Using the fact that $\lambda$ is a measure on $E$, show that for all $g\in L^+(X,M)$, $\int g(x)d\lambda (x)$=$\int f(x)g(x)d\mu (x)$

share|improve this question
    
Did you mean $\lambda(E)=\int_Ef(x)d\mu$? –  Asaf Karagila Oct 21 '11 at 21:01
    
You forgot $f$ in the definition of $\lambda$. What is $L^+$? –  AD. Oct 21 '11 at 21:02
    
You are absolutely right. $f$ is in the definition of $\lambda$. –  smanoos Oct 21 '11 at 22:22

1 Answer 1

up vote 5 down vote accepted

I guess $L^+(X,M,\mu)$ is the set of all nonnegative integrable function with respect to the measure $\mu$. Here is a hint to solve the problem:

  1. Show the result when $g$ is the characteristic function of a measurable set $M$.
  2. Using linear combination of such functions, show that the result holds when $g$ is a simple function.
  3. Use the fact that a nonnegative integrable function can be pointwise approximated by simple function, and conclude by the monotone convergence theorem.
share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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