I am currently trying to understand finitely additive (signed) measures, mostly reading Dunford & Schwarz and Rao & Rao. Knowing only the normal measure theory, the finitely additive measures seem to have quite some peculiarities. Hence my questions:
1.) The outer measure of a finitely additive (non-negative) measure on a field $\mathcal{A}$ of subsets of $X$ is a set function $\mu^{*}:\mathcal{P}(X)\rightarrow [0, \infty]$ defined by: $$ \mu^{*} = \inf\{\mu(B):A\subset B, B\in \mathcal{A}\}, \quad A \in X.$$ (Rao and Rao, Def. 4.1.3)
My questions: This does not resemble at all the definition of an outer measure. So why is the outer finitely additive measure defined like this?
2.) A null set is defined as $\mu^{*}(A)=0$ for $A\subset X$. Now a real valued function $f$ on $X$ is called a null function if: $$|\mu|^{*}(\{x\in X:|f(x)|<\varepsilon\})=0, \quad \forall\varepsilon>0, $$ where $|\mu|$ denotes the total variation. (Rao and Rao, Def. 4.2.4)
In Dunford & Schwarz, p.108, they write that a null-function need not vanish almost everywhere w.r.t. a finitely additive measure.
My question: What is this Definition good for then if it doesnt even vanish almost everywhere? Also, why doesnt it vanish a.e.? And why dont we define it in such a way then that it vanishes a.e.? (Bonus: Why do we use the total variation in the def?)
3.) It can be shown (Rao and Rao, Proposition 2.3.2 (i)) that a finitely additive measure on a field $\mathcal{A}$ is countably additive (on that field) if it is continuous from below (here the continuity property of measures is meant, i.e., when having an increasing set sequence, the measure of the sets converges to the measure of the union of these sets)
My questions:
(i) Can a measure that is defined on an algebra and not on a sigma algebra even be sigma additive? Since the right hand side of the equation $\sum_{i=1}^\infty \mu(A_i)=\mu(\cup A_i)$ wouldnt even be defined since the algebra is not closed under the union of countable sets, right?
(ii) Of what use is a sigma additive measure on an algebra? As I understand it, we have to use sigma algebra in measure theory precisley to make sense of the theory, so what happens if we now define that measure on an algebra?