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In this work, a mode of convergence for measurable functions is introduced. Arelated notion of Cauchy sequence is given and it is proved that this notion ofconvergence is complete in the sense that Cauchy sequences converge. Moreover,the preservation of convergence under composition is investigated. The originof this mode of convergence lies in the path of proving that the density of aEuler system converges almost everywhere (up to subsequences) towards thedensity of a non-linear diffusion system, as a consequence of the convergencein the relaxation limit.

A mode of convergence arising in diffusive relaxation