How can I extend a finitely additive measure?

Solution 1:

Yes!... But maybe No! :P

This is theorem $3.2.5$ in Rao and Rao's Theory of Charges: A Study of Finitely Additive Measures:

Let $\mu$ be a real partial charge on a collection $\mathscr{C}$ of subsets of a set $\Omega$. Let $\mathscr{F}$ be any field containing $\mathscr{C}$. Then there exists a real charge $\overline{\mu}$ on $\mathscr{F}$ which is an extension of $\mu$ from $\mathscr{C}$ to $\mathscr{F}$.

The authors use "charge" to mean a finitely additive measure, and this theorem is slightly more general than you need. A partial charge defined on a ring of sets is already a charge.

Unfortunately, the authors allow general charges to be signed, and their proof of this theorem crucially uses this fact. Even if $\mu$ is a positive charge, it's possible that the extension $\overline{\mu}$ will be signed. Alternatively, we can always find a positive extension $\overline{\mu}$, but we must allow it to take infinite values.

However, there are circumstances where we can guarantee a positive charge admits a finite positive extension. See, for instance theorem $3.2.9$:

Let $\mathscr{C}$ be any collection of subsets of a set $\Omega$, with $\Omega \in \mathscr{C}$. Let $\mu$ be a positive real partial charge on $\mathscr{C}$. Let $\mathscr{F}$ be any field on $\Omega$ containing $\mathscr{C}$. Then there is a positive charge $\overline{\mu}$ on $\mathscr{F}$ which is an extension of $\mu$.

The proofs are not hard, but they are somewhat long and technical, otherwise I would include them. The sketch is to see how to add a single set $A$ to $\mathscr{C}$ and choose what $\mu(A)$ should be in a way that's consistent. Then we (transfinitely) induct on all the sets in $\mathscr{F} \setminus \mathscr{C}$ in order to (eventually) get an extension on all of $\mathscr{F}$ that's consistent.

The authors actually describe lots of related theorems on how to extend charges off of a subalgebra, and it might be worth reading the whole chapter to get a result that's more tailored to your specific use case. The book is extremely readable, so you shouldn't have many issues flipping through it!

I hope this helps ^_^