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Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$?

For me, since the measure of each point $N$ is 0, therefore the Lebesgue integral evaluates to 0 i.e. $\mu(N) = 0$ $\forall N$. But isn't it counter intuitive since if we were to use the "area" intuition of the integral, no matter how large $N$ extends, the total area that we will get is always going to be 0? How does it make sense that area under the $f(n)$ consisting infinitely many bars over each $\mathbb{Z_+}$ be zero?

for example: $f(n)$

(Imagine those dots are filled)

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    $\begingroup$ I don't see why this is counterintuitive. None of those bars have any area. $\endgroup$ – Michael Albanese Jul 30 '15 at 21:35
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    $\begingroup$ If you sample, you probably don't want to integrate with respect to the Lebesgue measure. The counting measure would be a good candidate. $\endgroup$ – Daniel Fischer Jul 30 '15 at 22:14
  • $\begingroup$ $\sum_{n\in\mathbb Z} 0 = 0$... $\endgroup$ – Math1000 Jul 30 '15 at 22:45
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Any set of reals which is countable (i.e. finite or countably infinite) has zero Lebesgue measure. Lebesgue integration over a Lebesgue-null set always gives zero.

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