# How to extend a finitely additive measure

boolean-algebrameasure-theory

Let $$B$$ be a boolean algebra. Suppose we have a finitely additive measure $$\mu$$ defines on a subalgebra $$A\subseteq B$$.

Is it possible to extend $$\mu$$ to a finitely additive measure $$\nu$$ on
$$B$$? Is there a related theorem about extension of measures?

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 ^_^