What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

Question

1. What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?
2. What does it mean for $s$ to be integrable?

1. This is last minute exam revision. I really can't see what is going on here. It seems like we will be doing something like $\int_{\mathbb{N}} s d\mu = (1)s(1) + (2)s(2) + (3)s(3) +...$?

But I'm really not sure of that.

2. $s$ being integrable means that for every sub-interval we get when we partition the range of $s$, that sub-interval will be an element of $\mathbb{P}(\mathbb{N})$? Is that correct?

Answer 1

Notice that $\mathbb{N}$ is a disjoint union of singletons so:

$$\int_{\mathbb{N}}sd\mu=\sum_{k=1}^{\infty}\int_{\{k\}}sd\mu=\sum_{k=1}^{\infty}s(k)\mu(\{k\})=\sum_{k=1}^{\infty}s(k)$$

To be integrable means the above sum convergses absolutely.