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In this work we study the Lebesgue property for convex risk measures on thespace of bounded c\`adl\`ag random processes ($\mathcal{R}^\infty$). Lebesgueproperty has been defined for one period convex risk measures in \cite{Jo} andearlier had been studied in \cite{De} for coherent risk measures. We introduceand study the Lebesgue property for convex risk measures in the multi periodframework. We give presentation of all convex risk measures with Lebesgueproperty on bounded c\`adl\`ag processes. To do that we need to have a completedescription of compact sets of $\mathcal{A}^1$. The main mathematicalcontribution of this paper is the characterization of the compact sets of$\mathcal{A}^p$ (including $\mathcal{A}^1$). At the final part of this paper,we will solve the Capital Allocation Problem when we work with coherent riskmeasures.

Convex Risk Measures: Lebesgue Property on one Period and Multi Period Risk Measures and Application in Capital Allocation Problem