# The measures in Furstenberg's correspondence

In the paper Inverting the Furstenberg correspondence (IFC), the author defines a function $D_{A}(\sigma)$ on the Basic clopens of Cantor space, $2^{\mathbb{N}}$, where $A$ is a finite binary string of length $n$. Essentially $D_{A}(\sigma)$ counts the number of times that $\sigma$ is a substring of $A$ (where wraparound is allowed) and then divides that by $n$.

Via Caratheodory's extension theorem, $D_{A}$ is extended to a measure $\mu_{A}$ on all of $2^{\mathbb{N}}$. By weak$^{*}$ compactness of the space of measures, then, he obtains Theorem 2.1:

For every sequence $(A_n)$ of sets with $A_n \subseteq n$, there are a $T$-invariant measure $\mu$ on $2^{\mathbb{N}}$ and a subsequence $(A_{n_i})$ of $(A_n)$ with the property that for every $\sigma$, $\mu([\sigma]) = \lim_{i\to \infty} D_{A_{n_i}}(σ)$.

Now here is the statement I'm having trouble understanding:

We can take Theorem 2.1 to be a precise statement of the Furstenberg correspondence principle, though sometimes the phrase is used to refer to one of its consequences.

I've read Furstenberg's, Katznelson's, and Ornstein's paper The ergodic theoretical proof of Szemerédi's theorem (ETPST), in which, beginning with a set $\Lambda$ of positive upper density, they construct a sequence of measures $\mu_{n}$ that become "more and more $T$-invariant" and then take the desired $\mu$ to be any weak$^{*}$ limit of that sequence. (I'll add more detail here if necessary, but since the paper is freely available at the link I provided, all the details can be found on page 529 of that document.)

I'm not sure if the measures given in the two papers are related (or even somehow the same), but I don't think they're the same since the sequence of measures in IFC are shift invariant whereas those in ETPST aren't.

How is the correspondence described in IFC related to that in ETPST?

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Assuming that Avigad’s results are correct, the measure obtained in ETPSZ is necessarily obtainable for some sequence $\langle A_n:n\in\Bbb{N}\rangle$ of sets as in his Theorem 2.1: this is the essential content of Section 3, unless I’m badly misreading it. –  Brian M. Scott Mar 4 '12 at 11:55
@BrianM.Scott That's a good point. But I'm really interested in how Avigad interprets 2.1 as the content of p.529 of ETPST. I think I've basically figured it out though, with just a few details to iron out. –  Quinn Culver Mar 4 '12 at 17:44
As I said in the comment above, I've figured this out. I'll try to post an answer here someday. If you're reading this and want the answer, just email me. –  Quinn Culver Mar 23 '12 at 19:16