  This paper is concerned with the construction of atomic Gaussian
multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the
first hand, we construct purely atomic random measures corresponding to values
of the parameter $\gamma^2$ beyond the transition phase (i.e. $\gamma^2>2d$)
and check the duality relation with sub-critical Gaussian multiplicative chaos.
On the other hand, we give a simplified proof of the classical KPZ formula as
well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In
particular, this framework allows to construct singular Liouville measures and
to understand the duality relation in Liouville quantum gravity.
