Before revision: The objective of this paper is to present a comprehensive study of dual representations of risk measures and convex functionals defined on an Orlicz space L^\Phi or an Orlicz heart H^\Phi. The first part of our study is devoted to the Orlicz pair (L^\Phi,L^\Psi).
After revision: Let (\Phi,\Psi) be a conjugate pair of Orlicz functions. A set in the Orlicz space L^\Phi is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in L^\Phi characterizes closedness with respect to the topology \sigma (L^\Phi,L^\Psi).