The objective of this paper is to present a comprehensive study of the dual representation problem of risk measures and convex functionals on a Banach lattice $X$. Of particular interest is the case where $X$ is an Orlicz space or an Orlicz heart. The first part of our study is devoted to the pair $(X,X^\sim_n)$. In this setting, we present a thorough analysis of the relationship between order closedness of a convex set $C$ in $L_\Phi$ and the closedness of $C$ with respect to the topology $\sigma(L_\Phi(\mathbb{P}),L_\Psi(\mathbb{P}))$, culminating in the following surprising result: \emph{If an Orlicz function $\Phi$ and its conjugate $\Psi$ both fail the $\Delta_2$-condition, then there exists a coherent risk measure $\rho:L_{\Phi}(\mathbb{P})\rightarrow (-\infty,\infty]$ with the Fatou Property that does not admit a dual representation via $L_{\Psi}(\mathbb{P})$}. This result answers a long standing open problem in the theory of risk measures. In the second part of our study, we introduce the concept of the uo-continuous dual $X^{\sim}_{uo}$ and explore the representation problem for the pair $(X,X^\sim_{uo})$. This part extends the representation result for the pair $(L_{\Phi}(\mathbb{P}),H_{\Psi}(\mathbb{P}))$ established in [18] and complements the study of risk measures on an Orlicz heart $H_{\Phi}(\mathbb{P})$ developed in [8]. This paper contains new results and developments on the interplay between topology and order in Banach lattices that are of independent interest.
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