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)
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.