1. Can every integral (with respect to an integrable function) be written as a signed measure?
And does the function’s decomposition into positive and negative parts align somehow with the corresponding signed measure’s Jordan/Hahn decomposition into positive and negative measures?
2. Likewise, can every signed measure be written as the integral of some integrable function with respect to a measure (a la Radon-Nikodym)?
(Obviously this is trivially true for measures, just take the constant function $1$ as the integrand, but I want to know whether in general signed measures and integrals are in some sense equivalent concepts or not.)
3. Does some restriction to only functions of bounded variation somewhere come into play (even though we are considering arbitrary abstract measures)?
4. Would this lead to a quicker way of defining abstract Lebesgue(-Stieltjes) integrals?
I proved the first direction once, but I used the monotone and dominated convergence theorems, which seem to already require some notion of integration in order to be proved.
Perhaps there is a proof which does not a priori assume a notion of integral to be defined first. I.e. I would like to know if signed measures can really be shown to be the more fundamental object (and integrals as a convenient way of writing some subclass of them).
UPDATE: This problem is almost certainly related to the Jordan decomposition theorem for functions (which according to Wikipedia is something that exists). I will look into this further soon.