Let $\nu$ be a signed measure on $(X,\mathcal{M})$. We call the set $E$ positive if $\nu(F)\geq 0$ for any $F\subseteq E$. For a sequence $\{E_j\}_{j=1}^{\infty}$ of positive sets, it's easy to check that $\bigcup_{1}^{\infty}E_j$ is also a positive set. Intuitively, the sum of positive measure is still positive. However, given a sequence $\{G_i\}_{i\in I}$ of positive sets for uncountable index set $I$, do we have $\bigcup_{i\in I}G_i$ being a positive set?
Technically, I think it's not a positive set since the uncountable union of measurable set might not be in $\mathcal{M}$ anymore. Is my reasoning correct? However, I feel that the intuition "sum of positive measures should be positive" still holds here.
Hint by Noah Schweber, consider the signed measure $\nu(E)=\int_{E}fd\mu$ where $\mu$ is Lebesgue measure. Note that $E$ is a positive set if $f\geq 0$ almost everywhere. However, the uncountable union accumulates null sets into a set with a non-zero measure. That is, let $f=-1$ and $E=(0,1]$. Let $\{x_i\}_{x_i\in E}$ be a sequence of positive sets. However, $\nu(\bigcup_{i} {x_i})=\nu(E)=-1$ which is not a positive set.