  In this work we study the Lebesgue property for convex risk measures on the
space of bounded c\`adl\`ag random processes ($\mathcal{R}^\infty$). Lebesgue
property has been defined for one period convex risk measures in \cite{Jo} and
earlier had been studied in \cite{De} for coherent risk measures. We introduce
and study the Lebesgue property for convex risk measures in the multi period
framework. We give presentation of all convex risk measures with Lebesgue
property on bounded c\`adl\`ag processes. To do that we need to have a complete
description of compact sets of $\mathcal{A}^1$. The main mathematical
contribution 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 risk
measures.
