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In an article (see here) from the blog leisure of theory class, it is shown, in an interim stage for a Baysian game, we could run into a technical problem that the induced belief space for a player could be a uniform distribution on a countably infinite type space, which can't exist.

My question is how is it the case in that setting?

For example, let $a=\sqrt 2$, the nature choose $x=0$, for player 1, of course, $\sqrt{2} \mathbb Z$ is countably infinite, but how the induced distribution $p(\sqrt{2} \mathbb{Z} \mid x=0)$ is uniform? It seems to me it's trivial that $p((0, \sqrt{2}) \mid x=0) =p((0, 0) \mid x=0) = 0.5 $.

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