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Let $(B,\|\cdot\|)$ be a Banach space, $(\Omega,\mathcal{F},P)$ a probabilityspace and $L^0(\mathcal{F},B)$ the set of equivalence classes of strong randomelements (or strongly measurable functions) from $(\Omega,\mathcal{F},P)$ to$(B,\|\cdot\|)$. It is well known that $L^0(\mathcal{F},B)$ becomes a completerandom normed module, which has played an important role in the process ofapplications of random normed modules to the theory of Lebesgue--Bochnerfunction spaces and random functional analysis. Let $V$ be a closed convexsubset of $B$ and $L^0(\mathcal{F},V)$ the set of equivalence classes of strongrandom elements from $(\Omega,\mathcal{F},P)$ to $(B,\|\cdot\|)$, the centralpurpose of this paper is to prove the following two results: (1).$L^0(\mathcal{F},V)$ is $L^0$--convexly compact if and only if $V$ is weaklycompact; (2). $L^0(\mathcal{F},V)$ has random normal structure if $V$ is weaklycompact and has normal structure. As an application, a general random fixedpoint theorem for a strong random nonexpansive operator is given, whichgeneralizes and improves several well known results. We hope that our newmethod, namely skillfully combining measurable selection theorems, the theoryof random normed modules and Banach space techniques, can be applied in theother related aspects.

$L^0$--convex compactness and random normal structure in $L^0(\mathcal{F},B)$