# Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

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?

(Imagine those dots are filled)

• I don't see why this is counterintuitive. None of those bars have any area. – Michael Albanese Jul 30 '15 at 21:35
• If you sample, you probably don't want to integrate with respect to the Lebesgue measure. The counting measure would be a good candidate. – Daniel Fischer Jul 30 '15 at 22:14
• $\sum_{n\in\mathbb Z} 0 = 0$... – Math1000 Jul 30 '15 at 22:45