The set of possible kit-and-dart Penrose tilings is uncountably infinite. It would be very helpful to have some natural probability distribution $\mu$ over this set; such a distribution would allow us to make statements like "Viewed as an infinite planar graph, the expected degree of a Penrose tiling is...." and be a little more precise (see, e.g., papers like this).

To address some symmetry issues, let us restrict to the set of Penrose tilings with a tile corner at the origin and an edge between tiles along the positive x-axis ("an edge laying due east"). We will refer to this (also uncountably infinite) set as $\mathcal{P}$.

We would really like $\mu$ to cover *all* of $\mathcal{P}$ in some sense. So, does there exist a probability distribution $\mu$ over $\mathcal{P}$ that satisfies the following?

Consider $F$, an arbitrarily large, finite subset of tiles in the plane selected from any Penrose tiling in $\mathcal{P}$. Let $P_F\subseteq \mathcal{P}$ be the set of Penrose tilings that contains that fragment

in the same location. Then $\mu(P_F)>0$.

If there were an efficient algorithm for sampling from $\mu$, that would be even more helpful, but perhaps I'm hoping for too much.

Edit: Per Steven Stadnicki's and Mateusz Kwaśnicki's comment's, I've clarified that the fragment $F$ occurs in the same location.

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