Motivated by the recent work on conditional risk measures, this paper studies the Ekeland's variational principle for a proper, lower semicontinuous and lower bounded $\bar{ L}^{ 0}-$valued function, where $\bar{ L}^{ 0}$ is the set of equivalence classes of extended real-valued random variables on a probability space. First, we prove a general form of Ekeland's variational principle for such a function defined on a complete random metric space. Then, we give a more precise form of Ekeland's 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.
Categories: math.FA