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This is for anyone who has Rudin's Real and Complex analysis book at hand. I was looking at Rudin's Theorem 2.17 and 2.18. So far everything makes sense, except for one statement that Rudin makes in Theorem 2.18 on page 48.

He states the following: "Since $\lambda(K) < \infty$ for every compact $K$, $\Lambda$ is a positive linear functional on $C_c(X)$,...."

I'm more interested in how he used the hypothesis that $\lambda(K) < \infty$ for all compact set $K$ to conclude that $\Lambda$ is a positive linear function on $C_c(X)$.

At least to my limited knowledge, we are trying to verify that $\Lambda(\alpha f + \beta g) = \alpha \Lambda(f) + \beta \Lambda(g)$ for scalars $\alpha, \beta$ and $f,g \in C_c(X)$. In the first line of Theorem 2.18, Rudin sets $\Lambda f = \int_X f \ d\lambda$ for every $f \in C_c(X)$. Since each $f$ has compact support, then $ \Lambda f = \int_X f \ d\lambda = \int_K f \ d\lambda $. Then from there, I was uncertain how to verify $\Lambda$ is a positive linear functional.

Any tips for this question would be appreciated.

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    Go through the definition of "positive linear functional"; all the properties will follow from properties of the integral. Having $\lambda(K) < \infty$ just guarantees that $\int f\,d\lambda$ exists and is finite for $f \in C_c(X)$.2012-04-16
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    Just one question. I'm looking in the book, but I can't see how this is apparent: "$\lambda(K) < \infty$ guarantees that $\int_X f \ d\lambda$ exists and is finite for $f \in C_c(X)$." From this fact, if $f \geq 0$, I can deduce that $\Lambda$ is a positive linear functional on $C_c(X)$. Maybe there's a gap in my knowledge about this. I'll continue to look for this. Thanks though.2012-04-16
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    If necessary, you can go all the way back to the definition of the Lebesgue integral $\int f \,d\lambda$. But the key is that a continuous, compactly supported function must be bounded. So you are integrating a bounded function over a set of finite measure...2012-04-16
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    Thanks, Nate. I got it! ;)2012-04-16

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