Integration: Countable Additivity?

Question

When considering Bochner's theory of integration one notices that having a countable additive measure rather than merely additive measure is not important, or do I miss something?

Answer 1

An integral defined by integration against a finitely additive "measure" lacks the convergence theorems of Lebesgue integration. In particular it lacks the monotone convergence theorem. To see this, pick a disjoint family of sets $\{ A_k \}_{k=1}^\infty$, such that $\mu \left ( \bigcup_{k=1}^\infty A_k \right ) \neq \sum_{k=1}^\infty \mu(A_k)$. Then $f_n \equiv \sum_{k=1}^n \chi_{A_k}$ is a monotone sequence of measurable functions with pointwise limit $f = \chi_{\bigcup_{k=1}^\infty A_k}$, but

$$\int f_n d \mu = \sum_{k=1}^n \mu(A_k) \not \to \int f d \mu = \mu \left ( \bigcup_{k=1}^\infty A_k \right )$$