the possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk, we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. first, i will quickly describe the case of bernoulli convolutions, which can be seen as generalizations of the cantor middle third set, and then the case of random iterations of matrices in $\mathrm{sl}(2, \mathbb{r})$ acting on the real projective line, where the stationary measure is unique under certain conditions and is called the furstenberg measure. it had been conjectured that the furstenberg measure is always singular when the random walk has finite support. there have been several counter-examples, and the aim of the talk will be to describe that of bourgain, where the measure even has a very regular density. i will explain why the construction works for any simple lie group, using the work of boutonnet, ioana, and salehi golsefidy on local spectral gaps in simple lie groups.