short version: How do I take an expectation over a continuum of iid variables, i.e.:
$$\mathbb{E}_{\theta_{-i}} [U(\theta_i, \boldsymbol{\theta}_{-i})]$$ where $\theta_j$ is essentially a map $\theta: [0, 1] \mapsto [\underline{\theta}, \overline{\theta}]$, the variables $\theta_j$ are iid with cdf $F(\theta_j)$, and the expectation has to be taken over $\boldsymbol{\theta}_{-i}$, that is the realizations of $\theta$ for the whole interval $[0, 1]$ except for $i$?
long version:
The problem comes from economics, specifically a Bayesian incentive compatibility issue in a public good problem.
We have multiple agents. Each agent has a type $\theta_i \in \Theta = [\underline{\theta}, \overline{\theta}]$. Types are randomly assigned at the beginning of the problem, drawn independently according to some cumulative distribution function $F(\cdot)$. Agents only know their own type, $\theta_i$, but not the vector of other agents' types, $\boldsymbol{\theta}_{-i}$.
A "principal", which we may think of as a city council or other local authority, wants to build a public good that provides agents with utility according to their type:
$$U(\theta_i) = \theta_i x(\widehat{\theta_i}, \widehat{\boldsymbol{\theta}_{-i}}) - t_i(\widehat{\theta_i}, \widehat{\boldsymbol{\theta}_{-i}})$$
where $x(\cdot)$ is the "quantity of the public good" (think of it as "how many green spaces" or something), produced according to how much the agents "like" the good, and $t_i(\cdot)$ is how much agent $i$ has to pay for the public good (known as a "transfer" in economics). Here's the thing: $t_i$ is increasing in all arguments and agents can lie about their type, by announcing to the principal that they are of type $\widehat{\theta_i}$ while they are in fact of type $\theta_i$. The principal cannot verify the agents' claims.
If the principal just let the agents announce their type and plan accordingly, everyone would shun the public good so that they have to pay less, but then they would also not be able to enjoy the public good as much as they could have if they had announced a type that is just a bit higher. To avoid this, the principal can propose the agents a mechanism, that is a menu of options $\{x_i(\widehat{\boldsymbol{\theta}}), t_i(\widehat{\boldsymbol{\theta}})\}$ which we want to build so that we reach the best possible quantity of public good.
We want the mechanism to be Bayesian incentive compatible, that is every agent, knowing the distribution of types $F(\cdot)$ and ignoring other agents' strategic behavior, must find it better (higher utility) in expectation to announce their real type than any other type. In symbols:
$$ \theta_i = \underset{\widehat{\theta_i} \in \Theta}{\operatorname{argmax}} \mathbb{E}_{\theta_{-i}} [\theta_i x(\widehat{\theta_i}, \boldsymbol{\theta}_{-i}) - t_i(\widehat{\theta_i}, \boldsymbol{\theta}_{-i})] \quad \forall \theta_i$$
Now, when the number of agents is some finite $n$, taking expectations over a finite number of iid variables $\theta_{-i}$ is relatively straightforward:
$$ \mathbb{E}_{\theta_{-i}} [\theta_i x(\widehat{\theta_i}, \boldsymbol{\theta}_{-i}) - t_i(\widehat{\theta_i}, \boldsymbol{\theta}_{-i})] = \int_\Theta \int_\Theta \cdots \int_\Theta \theta_i x(\widehat{\theta_i}, \boldsymbol{\theta}_{-i}) - t_i(\widehat{\theta_i}, \boldsymbol{\theta}_{-i}) d F(\theta_1) \cdots dF(\theta_n) $$
and then work this out. But what if there is instead a continuum of agents defined over the interval $[0, 1]$? Do I need to make any further assumptions?