Several recent results in real algebraic and analytic geometry rely on arguments coming from motivic integration. These results concern blow-Nash maps, real singularities via the blow-analytic and the arc-analytic equivalences, and, more recently, inner-Lipschitz maps.

Similarly to the complex case, the real motivic measure assigns a "volume" to some sets of real analytic arcs on a real algebraic variety. However, it is not possible to use the complex construction as it is because Chevalley's theorem and the Nullstellensatz do not hold in this real context.

First I will explain the construction of this real motivic measure and how to process the cited above issues. Then I will focus on an application of this real motivic measure: an inverse mapping theorem for blow-Nash maps (i.e. semialgebraic maps which become real analytic after being composed with blowings-up). These maps recently played an important role in real algebraic geometry and in the classification of real singularities.