Before revision:  In this setting, we present a thorough analysis of the relationship between order closedness of a convex set \mathcal{C ] in L^\Phi and the closedness of \mathcal{C topology \sigma(L^\Phi,L^\Psi) , culminating in the following surprising result:If (and only if) an Orlicz function \Phi and its conjugate \Psi both fail the \Delta_2-condition, then there exists a coherent risk measure on L^\Phi that has the Fatou property but does not admit a dual representation via L^{\Psi . This result answers an open problem in the representation theory of risk measures. In the second part of our study,we explore the representation problem for the pair (H^\Phi,H^\Psi) . This part complements the study for the pair (H ^\Phi,L^\Psi) investigated in 8 and the study for the pair (L^{\Phi,H^{\Psi})investigated in } 19 . This paper contains new results and developments on the interplay between topology and order in Orlicz spaces that are of independent interest\emph{  .
After revision:  In this setting, we present a thorough analysis of the relationship between order closedness of a convex set \mathcal{C ] in L^\Phi and the closedness of \mathcal{C topology \sigma(L^\Phi,L^\Psi) , culminating in the following surprising result:If (and only if) an Orlicz function \Phi and its conjugate \Psi both fail the \Delta_2-condition, then there exists a coherent risk measure on L^\Phi that has the Fatou property but does not admit a dual representation via L^{\Psi ; that is, a convex set is \sigma(L^\Phi,L^\Psi)-closed if and only if it is closed with respect to the bounded-\sigma(L^\Phi,L^\Psi) topology. As a result, we show that order closedness and \sigma(L ^\Phi,L^\Psi) investigated in 8 and the study for the pair (L^{\Phi,H^{\Psi})investigated in } 19 . This paper contains new results and developments on the interplay between topology and order in Orlicz spaces that are of independent interest\emph{ .