In spite of a large amount of research on the problem of fluid mixing and its control, there is no consensus on a proper measure for quantifying mixing. We present a measure of mixing that is based on the concept of weak convergence and is capable of probing the "mixedness" at various scales. This new measure, called the mix-norm, resolves the inability of the L/sub 2/ variance of the scalar density field to resolve various stages of contour-level rearrangement by chaotic maps. In addition, the mix-norm succeeds in capturing the efficiency of a mixing protocol in the context of a particular initial density field, wherein Lyapunov-exponent based measures fail to do so. The mix-norm is a pseudo-norm for checking weak convergence on the space of scalar density fields, which turns out to be a critical link in justifying its validity as a measure for mixing. We demonstrate the utility of the mix-norm by showing how it measures the efficiency of mixing due to diffusion and to various discrete dynamical systems.