Let $(X,\mathcal{A},\nu)$ be a signed measure space such that $\nu(E)\in [-\infty, +\infty)$ for all $E\in\mathcal{A},$ and suppose $$\sup\{\nu(E)\;|\; E\in\mathcal{A}\}=+\infty.$$
- Does there exist a sequence $\{E_n\}\subseteq\mathcal{A}$ such that $\lim_{n\to\infty}\nu(E_n)=+\infty$?
- If so, can it be chosen such that moreover, $\nu(E_n)\in\mathbb{R}$ for all $n\in\mathbb{N}$?
My attempt at answering question number 1.
We have for definition of infinity sup that $$\forall m\in\mathbb{N}\quad\exists E_m\in\mathcal{A}\quad\text{such that}\quad \nu(E_m)> m.\tag1$$ Let be $M>0$, then exists $m\in\mathbb{N}$ such that $m>M$ and therefore from $(1)$ exists $E_m\in\mathcal{A}$ such that $\nu(E_m)>M$. Thus we have proved that $$\forall M>0,\quad \nu(E_n)>M\quad \forall n\ge m$$ that is $\nu(E_n)\in\mathbb{R}$ for all $n\ge m$ and $\lim_{n\to\infty}\nu(E_n)=+\infty$
