I will give a brief introduction to the theory of motivic integration developed by R. Cluckers and F. Loeser, which is based on model theory techniques, and allows to understand the measure on $p$-adic fields in a "uniform in $p$" way. I will also try to sketch some applications in representation theory, which can be thought of as a far-reaching extension of the Ax-Kochen-Ersov principle.