What is the First order logic representation of a measure space?

Question

How can a measure space be written in first order language? I'm presuming that it must be many sorted. How in particular can the measure be written?

Answer 1

Let us consider what we would need for such a construction, to define a measure space $(X,\Sigma,\mu)$.

Clearly, we need:

• sorts for $X, \Sigma, \overline{\Bbb R}_{\ge 0}$;
• a relation $\in: X \times \Sigma$, a function $\mu: \Sigma \to \overline{\Bbb R}_{\ge 0}$

But in order to specify that $\Sigma$ is closed under countable union, we cannot simply use a function symbol.

So we would need a sort for something like $P^{<\omega}\Sigma$, the set of countable subsets of $\Sigma$.

But this now requires to specify as an axiom that $P^{<\omega}\Sigma$ contains all countable subsets of $\Sigma$. So we need to talk about countability. But this brings other problems, like defining bijections, etc. etc.. We haven't even started to try and axiomatise the very nice behaviour of the positive extended real line $\overline{\Bbb R}_{\ge 0}$.

As one can see, this quickly grows out of hand, and the complication encountered makes using a suitable form of set theory as the language for measure theory a reasonable and attractive option -- if the desire to ditch a first-order logical approach and just reason in natural language instead hasn't taken over yet.