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 .
After revision:  In this setting, we present a thorough analysis of the relationship between order closedness of a convex set \mathcal{C ].) In this paper, we show that for a norm bounded convex set in L^\Phi, order closedness and \sigma(L^\Phi,L^\Psi)-closedness are indeed equivalent. In general, however, coincidence of order closedness and \sigma(L^\Phi,L^\Psi)-closedness of convex sets 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 .