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