We study Kusuoka representations of law invariant coherent risk measures on the space of bounded random variables. We refine this representation by giving that any law invariant coherent risk measure can be written as an integral of the Average Value-at-Risk measures on $[0,1]$, which gives a numerically constructive way to approximate any law invariant coherent risk measure. The results are illustrated on specific law invariant coherent risk measures along with numerical simulations.