Let $T$ be a complete theory (say in a countable language, though this may not be required). Assume that $T$ is complete with infinite models. 


Question 1) When is the theory of $T$ an almost sure theory? What are the references for these results.


Note that I have left the term "almost sure" undefined. This is because the notion seems to change from author to author. Some authors take it to mean the limit under uniform measures, others (The Shelah-Spencer almost sure theories for example) use more complicated measures. I have no restriction in mind and would like to know what results are out there for "reasonable" notions of almost sure theories.


Question 2) What are some good sources that contain results about pseudofinite model theory and the "interaction" with almost sure theories? 


Edit 1) The following should be added to the end of question 2: For example, assume that $T$ is pseudofinite. When is it an almost sure theory?



When are pseudofinite theories almost sure theories?
In this answer, let's only think about theories in countable relational languages. 


Most of the time, when people say that $T$ is an "almost-sure theory", they mean it arises as the limit theory for some sequence of probabilistic constructions which admits a zero-one law. We could formalize this by asking for a sequence of probability measures $(\mu\_n)\_{n\in\omega}$ on the space of all finite $L$-structures (let's say $L$-structures whose domains are initial segments of $\omega$, to make this a reasonable space), such that for every sentence $\varphi\in T$, $\lim\_{n\to\infty} \mu\_n([\varphi]) = 1$, where $[\varphi]$ is the set of all models of $\varphi$ in our space.


But this definition is probably too broad. Indeed, under this definition *every* pseudofinite theory is an almost-sure theory. Just enumerate $T$ as $\{\varphi\_i\mid i\in \omega\}$, let $M\_n$ be a finite model of $\bigwedge\_{i\leq n}\varphi\_i$, and let $\mu\_n$ be the Dirac measure concentrated on the point $M\_n$.


So what are some reasonable ways to refine the definition? 


One very minor thing we could do is to require the measure $\mu\_n$ to be a measure not on the space of all finite $L$-structures, but on the space $\mathrm{Str}\_L([n])$ of $L$-structures with domain $[n] = \{0,\dots,n-1\}$. (This is the case for the usual sorts of zero-one law examples like the uniform measures and the Shelah-Spencer measures).


By a similar construction to the one above, you can show that a pseudofinite theory $T$ is almost-sure for this modified notion if and only if for every sentence $\psi$ such that $T\models \psi$, the exists some $N\in \omega$ such that $\psi$ has a model of size $n$ for all $N\leq n$. This is a somewhat reasonable notion - it rules out theories whose finite models are special enough that they only show up sporadically. But it still allows us to use Dirac measures, which feels pretty unnatural.


Here are some other possible conditions you could put on the $\mu\_n$.


1. For any formula $\varphi(\overline{x})$ and any tuple $\overline{a}$ from $[n]$, let $[\varphi(\overline{a})] = \{M\in \mathrm{Str}\_L([n])\mid M\models \varphi(\overline{a})\}$. Require that for all $\varphi(\overline{x})$ quantifier-free, all $\overline{a}$ from $[n]$, and all $n\leq m$, $\mu\_n([\varphi(\overline{a})]) = \mu\_m([\varphi(\overline{a})])$. Weaker, you could only require that $\lim\_{n\leq m\to \infty} \mu\_m([\varphi(\overline{a})])$ converges.
2. The space $\mathrm{Str}\_L([n])$ comes with a natural action of the symmetric group $S\_n$ by permutation of the domain. Require $\mu\_n$ to be invariant under this group action. This is the same as requiring that $\mu\_n([\varphi(\overline{a})]) = \mu\_n([\varphi(\sigma(\overline{a}))])$ for all $\sigma\in S\_n$.
3. Require that whenever $\varphi(\overline{x})$ and $\psi(\overline{y})$ are quantifier-free formulas and $\overline{a}$ and $\overline{b}$ are disjoint tuples from $[n]$, $\mu\_n([\varphi(\overline{a})\land \psi(\overline{b})]) = \mu\_n([\varphi(\overline{a})])\mu\_n([\psi(\overline{b})])$. Or just require that this identity is true in the limit as $n\to \infty$.


If we assume condition 1, we get that the $\mu\_n$ converge in a natural way to a measure $\mu$ on $\mathrm{Str}\_L(\omega)$, the space of countable labeled $L$-structures. If we assume condition 2, we get that $\mu$ is invariant under the group action of $S\_\infty$ (the permutation group of $\omega$), and if we assume condition 3, it turns out that $\mu$ is ergodic for this group action. I think that these sorts of measures are very interesting, and I'm currently working on the relationship between these limit measures and the limit theory of the $\mu\_n$ (under additional hypothesis, i.e. that the limit theory is the theory of a Fraïssé limit and the limit measure $\mu$ gives measure $1$ to the isomorphism class of that Fraïssé limit!). 


On the other hand, I certainly don't want to suggest that a general notion of "almost sure theory" should include even condition 1, since condition 1 is not satisfied in general by the uniform measures on Fraïssé classes.


