We continue developing basic properties of the pcf function.

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this talk, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this talk, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Morley introduced the notion of a scattered sentence of $L_{\omega_1,\omega}$.  Roughly speaking, $\varphi$ is scattered if it does not have a perfect set of countable models.  He then showed that scattered sentences have at most $\aleph_1$ many countable models (up to isomorphism) whilst non-scattered sentences have continuum many nonisomorphic countable models.  The absolute Vaught conjecture states that scattered sentences have only countably many countable models up to isomorphism.  Unlike the original Vaught conjecture (which holds trivially under CH), the absolute Vaught conjecture does not depend on the model of set theory in question and is in fact equivalent to the original Vaught conjecture under the negation of CH.

Keisler connected the notion of scattered sentences with randomizations of structures.  Randomizations are models of a certain continuous theory, called the pure randomization theory, and in such a randomization, one can define the probability that $\varphi$ holds.  A randomization of $\varphi$ is a randomization in which $\varphi$ holds with probability one.  An example of a randomization of $\varphi$ is a basic randomization of $\varphi$, which consists of a collection of "measurable" random variables taking values in a countable family of models of $\varphi$.  $\varphi$ is is said to have few separable randomizations if every randomization of $\varphi$ is a basic randomization of $\varphi$.   

Keisler showed that if $\varphi$ has few separable randomizations, then $\varphi$ is scattered.  Moreover, he showed that, assuming Martin's axiom for $\aleph_1$, the converse holds.  Andrews, Goldbring, Hachtman, Keisler, and Marker were able to remove the use of Martin's axiom by an absoluteness argument.  Thus, it is a theorem of ZFC that $\varphi$ is scattered if and only if $\varphi$ has few separable randomizations.

In this talk, I will try to define most of the above results in more detail and sketch the ideas behind the proofs of the theorems alluded to above.

Recent years have seen an increasing number of applications of descriptive set theory in ergodic theory and dynamical systems. We present some set theoretic background and survey some of the applications.

Slides for this series of talks can be found here:

https://www.dropbox.com/sh/om8efuv6ez10ysb/AADOA4SPbdjXKoDajEftFb2pa?dl=0