pointwise limit of finite measures

Solution 1:

Take $\mu_n: \mathbb{N} \to \mathbb{R}$ to be $$ \mu_n(A) = \#(A \cap [n,\infty)). $$ For any bounded set $B$, you have that $\mu_n(B) \rightarrow 0$. But, for any unbounded set $U$, $\mu_n(U) = \infty \rightarrow \infty$.

The limit $\mu$ is not a measure, because $$ \mu(\mathbb{N}) = \infty \neq 0 = \sum_{n \in \mathbb{N}} \mu(\{n\}). $$

This example is quite easy to extend to $\mu_n(A) = \lambda(A \setminus K_n)$, where $K_n$ is a sequence of sets such that $K_n \uparrow K$, with $\infty \neq \lambda(K_n) \rightarrow \lambda(K) = \infty$.

In this case, $\mu(K) = \infty$, while $\mu(K_n) = 0$.