Hi there

I don't know how to solve this exercise:

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Let $\displaystyle \mu, \lambda $ be two measures on the borel set with $\displaystyle \lambda << \mu $ (this means that $\displaystyle \lambda$ is absolute continuous related to $\displaystyle \mu$). Prove the following statement or the contrary:

If $\displaystyle \mu$ is finite then $\displaystyle \lambda$ is finite.

""""

I'd guess that the statement is true. What's clear is that according to Radon-Nikodym there exists a measurable f such that:

$\displaystyle \lambda(A)=\int_A f d\mu $ and the assumption is $\displaystyle \mu(\mathbb{R})<\infty$

So now: $\displaystyle \lambda(\mathbb{R})=\int_{\mathbb{R}} f d\mu=...$

How can I complete this? Is there an upperbound for f I can take (a real number c)? If yes what theorem says that there is one? Then this proof was easy.

$\displaystyle \int_{\mathbb{R}} f d\mu \le \int_{\mathbb{R}} c d\mu = c*\mu(\mathbb{R})$

But I can't find a theorem or a corollary that says that. How can I show this?

Regards