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Rudin PMA p.130 Theorem 6.16

The theorem on this page states;


Let $\{c_n\}$ be a sequence of nonnegative reals and $\sum c_n$ be convergent. Let $\{s_b\}$ be a sequence of 'distinct' points in (a,b). Let $\alpha(x) = \sum_{i=1}^{\infty} c_n I(x-s_n), \forall x\in [a,b]$ (where $I$ is the unit step function)

If $f$ is continuous on $[a,b]$, then $\int_a^b f d\alpha = \sum_{i=1}^{\infty} c_n f(s_n)$.


I don't know where in the proof used the hypothesis '$\{s_n\}$ is a sequence of distinct points'. It seems to me that this hypothesis is not essential. (That is, it doesn't need to be distinct) Where did i go wrong?

share|improve this question
    
It isn't essential, you are right, but you can of course assume that the $(s_n)$ are distinct, by grouping correspoing $c_n$'s. –  martini Nov 26 '12 at 14:12
    
related: Question about integral and unit step function –  draks ... Nov 26 '12 at 14:14
    
@martini, draks The link helps me understanding this a lot. Thank you –  Katlus Nov 26 '12 at 14:38

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