I would like to ask whether there is some kind of analogue of outer measure when dealing with signed measures. I would like to assign measure to all the subsets, not just some $\sigma$-field. I'm using Evans-Gariepy's book "Measure theory and fine properties of functions" and I would like to apply their approach/theory to signed measures.
I also know that you can define an outer measure given a non-negative measure $\mu$ on a given $\sigma$-field $(X, \sigma)$ by defining $$ \mu'(A) := \inf\left\{\sum_{i \in I} \mu(B_i) | B_i \in \sigma, i\in I\ {\rm countable}, \ A \subset \bigcup_{i \in I} B_i\right\} $$
for ANY subsetset $A$ of the underlying set $X$ (so we allow measuring of non-measurable sets in Caratheodory sense). Such an outer measure makes members of $\sigma$ into $\mu'$-measurable sets and $\mu'(B) = \mu(B)$ for all $B \in \sigma$ just like we would like it to.
But this approach doesn't seem to work for signed measures.
Maybe Jordan decomposition could work?
If you know the book I'm talking about, I will be very thankful for your advice and help.
