Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitly additive measure. If $f : X \to [0,\infty]$ is a measureable function, we can define $$ \int_{X}f\,d\nu$$ in the standard way. If $f,g :X \to [0,\infty]$ are simple measurable functions then it is easy to prove that $$\int f + g\,d\nu = \int f\,d\nu + \int g\,d\nu. $$ However, if $f$ and $g$ are just measurable functions, then it is only obvious that $$ \int f\,d\nu + \int g\,d\nu \leq \int f + g\, d\nu. $$

Question : Does integration with respect to a finitly additive measure respect addition?

Note, that if $\nu$ is countably additive, then the standard way to prove that integration respects addition is to appeal to the monotone convergence theorem.

Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we can define $$ \int_{X}f\,d\nu$$ in the standard way. If $f,g :X \to [0,\infty]$ are simple measurable functions then it is easy to prove that $$\int f + g\,d\nu = \int f\,d\nu + \int g\,d\nu. $$ However, if $f$ and $g$ are just measurable functions, then it is only obvious that $$ \int f\,d\nu + \int g\,d\nu \leq \int f + g\, d\nu. $$

Question : Does integration with respect to a finitely additive measure respect addition?

Note, that if $\nu$ is countably additive, then the standard way to prove that integration respects addition is to appeal to the monotone convergence theorem.