  Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N.
Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct.
Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of
conditional risk measures. Later, the algebra of conditional sets was
introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra
of conditional sets and the concepts of conditional topology and compactness.
J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally
$L^0$-convex modules, and find exactly which subclass of locally $L^0$-convex
modules can be identified with the class of locally convex vector spaces within
the context of conditional set theory. Second, we provide a version of the
classical James' theorem of characterization of weak compactness for
conditional Banach spaces. Finally, we state a conditional version of the Fatou
and Lebesgue properties for conditional convex risk measures and, as
application of the developed theory, we stablish a version of the so-called
Jouini-Schachermayer-Touzi theorem for robust representation of conditional
convex risk measures defined on a $L^\infty$-type module.
