Confusion about proving there is no finite measure on a countable set $X$ other than the trivial measure which assigns measure $0$ to every set.

Question

A measure $\nu$ is finite if $\nu(X) < \infty$.

Let $X$ be a countable set and let $\mathcal A$ be the $\sigma$-algebra of all subsets of $X$. Prove there is no finite measure on $X$ other than the trivial measure which assigns measure $0$ to every set.

What if $X$ is finite? We can assign to each set in the $\sigma$-algebra, the cardinality of the set. So $\nu(X)=|X|$.

Answer 1

The claim is not true. Let $(x_k)_{k=1}^{\infty}$ be an enumeration of $X$. Then let $\mu(\{x_k\}) = 2^{-k}$. This generates a finite measure on $X$ with $\mu(X) = \sum_{k=1}^{\infty} 2^{-k} = 1$.