I originally posted this on math.stack, but no one answered, so im posting here:

I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for actions of amenable groups"):

Begin by taking the direct product of two dyadic odometers. The space for this $\mathbb{Z^2}$-action can be thought of as the collection of all the different ways of organizing $\mathbb{Z^2}$, first into a tiling by

$2$ x $2$ squares, then a tiling by $4$ x $4$ squares, each well tiled by the $2$ x $2$ squares, then a tiling by $8$ x $8$ squares, etc. $\mathbb{Z^2}$ acts on this space of patterns by translations. Next to each such pattern we will associate to each element of $\mathbb{Z^2}$ a random variable taking the values $+-$ with probability $(1/2,1/2)$. These random variables will be independent, subject to the following constraints:

$(1)$ in every $2$ x $2$ square the bottom pair is equal to the top pair;

$(2)$ in every $4$ x $4$ square, the left pair of $2$ x $2$ squares equals the right pair of $2$ x $2$ squares;

$(3)$ in every $2^3$ x $2^3$ square the bottom pair of $2^2$ x $2^2$ squares equals the top pair of $2^2$ x $2^2$ squares;

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$(2n)$ in every $2^{2n}$ x $2^{2n}$ square the left half of the square equals the the right half of the square;

$(2n+1)$ in every $2^{2n+1}$ x $2^{2n+1}$ square the bottom half of the square equals the top half of the square.

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It is clear that along the $45^∘$ diagonals one sees independent random variables, thus the entropy of $ST$ is at least $\log 2$, while in the vertical and horizontal directions the entropy is killed because we repeat exactly, infinitely many times. Thus $h(S)= h(T)=0$ but $h(ST)≥\log2$.

I don't understand what exactly is the measure space in this construction. Is the underlying set a subset of $\mathrm{odometer}$ x $\mathrm{odometer}$ x $\{+,-\}^{\mathbb{Z^2}}$$?$ What is the measure?

Can someone help me figure it out? it's probably standard.

Thanks.

Here is a picture of the construction, if you prefer.