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 measure* means 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.



"Compactness in Measure" in Function Spaces
Probably the authors refer to the space $L\_0(\mu)$ of all (equivalence classes) of measurable functions. This is a complete metric space in the metric I have mentioned in a comment above or, which is closer to the quotation in the question, for the equivalent metric given by $d'(f,g)=\inf\_{s>0} \{s+\mu\{|f-g|\ge s\}\}$, i.e., $d'(f,g)=\|f-g\|$ in the notation of the question. But note that $\|\,.\,\|$ is not a norm; $L\_0(\mu)$ is a non-locally convex complete metric space. This topology on $L\_0$ is called topology of convergence in measure, and one uses the epithet "in measure" to refer to this topology, hence the phrases "closed in measure" or "compact in measure". A fundamental reference for sets closed in measure is A.V. Bukhvalov, G.Ya. Lozanovskij, On sets closed in measure in spaces of measurable functions. Trans. Mosc. Math. Soc. 34, 127-148 (1978). As for sets compact in measure see G. Godefroy, N. Kalton, D. Li, On subspaces of $L\_1$ which embed into $\ell\_1$. J. Reine Angew. Math. 471, 43-75 (1996). 


