In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". They say

Here

compactness in measuremeans compactness in the normed space $S$ of all measurable, almost everywhere finite functions $x$, equipped with the norm$$||x|| = \inf_{s>0} \{s + \text{mes} \{t \, : \, |x(t)| \geq s\}\}$$

Where "$\text{mes} \, D$" means the measure of the set $D$.

My questions are: Does this property go by any other names? And are there good sources in English which mention it?

The only sources I can find which use it are papers by N. A. Erzakova, not all of which have been translated into English, and possibly a paper in Russian by P. P. Zabreiko.