# Is maximal utility conditional on information linear in convex combinations of priors?

This is related to a Mathematica question here - https://math.stackexchange.com/q/1952779/374929

Is a (maximal expected utility) function of the form

$U(\mu, X) \equiv \int_\Theta \int_\mathcal{X} \max_a \int_\Theta u(a, \theta) d \mu(\theta|x) d P_\theta (x) d \mu(\theta)$

linear in convex combination of priors; that is, is it true that

$U(\alpha \mu + (1-\alpha) \nu, X) = \alpha U(\mu, X) + (1-\alpha) U(\nu, X)$?