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.

中文翻译：

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扩散弛豫中产生的收敛模式

在这项工作中，引入了可测量函数的收敛模式。给出了柯西序列的相关概念，并证明了该收敛概念在柯西序列收敛的意义上是完备的。此外，还研究了合成下收敛性的保持。这种收敛模式的起源在于证明欧拉系统的密度几乎在所有地方（直到子序列）都向非线性扩散系统的密度收敛，这是松弛极限收敛的结果。