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In McCallum's, "Monetary Economics", it is implied that $I_t$ is a set that contains all information $\{x_t,x_{t-1},..., y_t,y_{t-1}, ..., u_t, u_{t-1}, ...\}$, where $x_t$ is the value of variable $x$ for time $t$. This means that an information set contains all the known variable prices, up to period $t$ and prior, including knowledge over ...

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Here are my thoughts. Because the sender commits to a disclosure policy, how you stipulate off-path beliefs doesn't matter (phrased loosely). That is, the modification to PBE is trivial: any optimal persuasion policy corresponds to a class of optimal persuasion policies in which any belief is assigned to each off-path message. Moreover, you could also ...

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Two points. Common knowledge is defined by Aumann in terms of partitions, not $\sigma$-algebras. There is generally no natural correspondence between these. The grand state space is trivially common knowledge. So however you conceive of the relevant states, something that holds everywhere, such as the structure of the model, is common knowledge even if ...

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