I was wondering how you could prove the linearity of the integral without using that measures are $\sigma$-additive. I have no clue of where to start, but let me state my question more precisely. Suppose $\mathfrak{M}$ is a $\sigma$-algebra on a set $\Omega$. If I would redefine measures in such a way that it suffices for a function $\mu: \mathfrak{M} \to [0, + \infty]$ to be finitely additive in order for it to be a measure, how can I than prove that the integral of positive measurable functions on a set is linear? The definition of the integral of positive measurable goes as follows of course: $$\int_\Omega f d \mu = \sup \{ \int_\Omega s d \mu | \text{ $s$ is a positive measurable step function with $s \leq f$} \}.$$ (I guess the definiton of step functions and their integrals is known by most people who cared to read this far)
Thanks!
Disclaimer: I'm not suggesting that the $\sigma$-additivity of measures is not a necessary part of a satisfactory theory of integration. For instance the monotone convergence theorem -a result we all desperately want to be true- hinges on measures being $\sigma$-additive.