In spite of a large amount of recent 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{sup}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.
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