There is an atomless probability space $(\mathbb{I},\mathfrak{I},\eta)$ representing a set of agents indexed by $i\in[0,1]$, such that the probability space is the Lebesgue unit interval. Each type receives a payoff $x_{i}$ with has a distribution $F(\mu,\sigma)$, being continuous with a support $[\underline{x},\bar{x}]<\infty$. Is the following correct?
$E[\int_{0}^{1}x_{i}~1_{\{x_{i}\geq \hat{x}\}}~di]=\int_{\underline{x}}^{\bar{x}}\int_{0}^{1}x_{i}~1_{\{x_{i}\geq \hat{x}\}}~di~f(x)~dx=\int_{0}^{1}\int_{\underline{x}}^{\bar{x}}x_{i}~1_{\{x_{i}\geq \hat{x}\}}~f(x)~dx~di=\alpha\mu$
where $\hat{x}$ is some real value and I define $\alpha\equiv\int_{0}^{1}1_{\{x_{i}\geq \hat{x}\}}~di$.
The above is correct only if Fubini extension holds. These are my references: Sun, 2006 and Sun, 2009