Is this measure sigma-finite?

Question

I have a really easy question. Is this measure:

$\mu(E)= \sum_{n\in{E}}(n+1)$

sigma finite on $\ (N,P(N))$ ?

I am not sure since if I take $E_n={n}$ then $\mu(E_{n})$ is not finite for all n isn't it?

thank you in advance

Answer 1

$\mu \{1,2,...,N\}=2+3+...+(N+1) <\infty$ for all $N$ and $\bigcup_N \{1,2,...,N\}=\mathbb N$ so the measure is sigma finite.