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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 $g\in L^+(X,M)$, $\int g(x)d\lambda (x)$=$\int f(x)g(x)d\mu (x)$

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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
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.
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