Proving Absolute Convergence of a Real-Valued Function on a Sigma Algebra

[Fermat1]

Let R be a sigma algebra and let $f$ be a real value function on R such that for a sequence
($A_{n}$) of disjoint members of R, we have that the sum of $f$($A_{n}$) over all n is equal to the image of the countable union under $f$. Prove that the sum of $f$($A_{n}$) is in fact absolutely convergent.

 

[Jhoneine]

Since $f$ is a real-valued function, it must be bounded. That is, there exists a constant M such that |$f$($A_{n}$)| $\leq$ M for all n. By the Principle of Finite Sums, we know that the sum of a sequence of finitely many non-negative real numbers is convergent if and only if the sequence of partial sums is increasing and bounded above. Since the sum of $f$($A_{n}$) over all n is equal to the image of the countable union under $f$, the sequence of partial sums is increasing and bounded above by M. Therefore, the sum of $f$($A_{n}$) is absolutely convergent.