In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a classical weak solution of the Euler equations. This is a joint work with Gui Qiang Chen (Oxford University and Northwestern University).