First I will briefly recall the definitions of the objects introduced in my previous talk together with the construction of the motivic measure in

the real algebraic context.

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 finitely many blowings-up). These maps recently played an important role in real algebraic geometry

and in the construction of a discrete classification of real singularities.