# Convergence of a sequence to a $\sup$ in a signed measure space.

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 $$\sup\{\nu(E)\;|\; E\in\mathcal{A}\}=+\infty.$$

1. Does there exist a sequence $$\{E_n\}\subseteq\mathcal{A}$$ such that $$\lim_{n\to\infty}\nu(E_n)=+\infty$$?
2. 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 $$\lim_{n\to\infty}\nu(E_n)=+\infty$$

• Commented Sep 18, 2023 at 16:16
• @AnneBauvalExcuse me. I edited by posting my solution to question number 1. Commented Sep 18, 2023 at 17:06
• You also solved question 2. Note that it has nothing to do with measure theory. Simply: the sup of any nonempty $S\subset[-\infty,+\infty]$ is the limit of a sequence in $S.$ en.wikipedia.org/wiki/Infimum_and_supremum#Properties Commented Sep 18, 2023 at 17:27