Given a sequence $(\mu_{n})_{n\in\mathbb{N}}$ of $\sigma$-finite measures on the measurable space $(Ω,\Sigma)$, is the $\limsup_{n}\mu_{n}$ also a $\sigma$-finite measure?
Clearly, $\limsup_{n}\mu_{n}(A)\geq 0$ for all $A\in \Sigma$, and that $\limsup_{n}\mu_{n}(\varnothing)=0$. But I am not sure if $\sigma$-additivity holds.
I have read about the Vitali-Hahn-Saks theorem which shows that the limit of a sequence of measures is also a measure. Can we adapt the proof of Vitali-Hahn-Saks' theorem or is there a counterexample?