Ekelandʼs variational principle for an -valued function on a complete random metric space

Motivated by the recent work on conditional risk measures, this paper studies the Ekelandʼs variational principle for a proper, lower semicontinuous and lower bounded -valued function, where is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekelandʼs variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekelandʼs variational principle for such a local function on a complete random normed module. Finally, as applications, we establish the Bishop–Phelps theorem in a complete random normed module under the framework of random conjugate spaces.