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In the usual setup for the competitive equilibrium we have the agent solve:

$$ \max_{\{c_i(h^t)\}} \sum_{t=0}^\infty \sum_{h^t} \beta ^t u(c(h^t)) \pi(h^t) $$

subject to constraints, where $h^t$ denotes the history at time $t$, $\beta$ is the discount factor and $c(h^t)$ is history-state consumption.

My question is, what do we do when the underlying state each period is continuous? Is it well defined to write,

$$ \max_{\{c_i(h^t)\}} \sum_{t=0}^\infty \int \beta ^t u(c(h^t)) \pi(h^t) \: dh^t, $$

or is there some condition we need to make it well defined? In the RBC model for instance, everyone just writes an expectation over the continuous possible realisations of $A_t$ and it is never written explicitly.

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