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How can you prove that if $\mu$ and $\nu$ are finite measures and $n$ is a positive real number, then $\mu$+$n\nu$ is again a finite measure? Is that the same for $\sigma$-finite measures?

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Are you sure that $n$ could be any real number? –  Stefan Hansen Nov 25 '12 at 17:18
    
I'm confused: if you are given that $\mu (X) = K$ and $\nu (X) = K'$ then $\mu(X) + n \nu (X) = K + n K'< \infty$. What am I missing? –  Rudy the Reindeer Nov 25 '12 at 17:23
    
And you probably mean $n \in \mathbb R_{\ge 0}$. –  Rudy the Reindeer Nov 25 '12 at 17:24
    
@MattN. they could be signed measures, I suppose. Wikipedia/ba space. –  kahen Nov 25 '12 at 17:30
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What is/are the axiom/axioms of a measure you have problems to check? –  Did Nov 25 '12 at 17:43

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