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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

$$\|x\| = \inf_{s>0} \big\{s + \text{mes} \{t \, : \, |x(t)| \geq s\}\big\}$$

Where "$\operatorname{mes} D$" 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.

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  • $\begingroup$ Is the inf supposed to be over $s$? Are we working on any particular measure space, or a general one? I guess it has to be an infinite measure space? $\endgroup$
    – Nate Eldredge
    Commented Nov 27, 2018 at 15:36
  • $\begingroup$ Yes, the inf is over $s$; I've fixed that. This topic comes up in the section on $L^p(V)$, where $V$ is a set of finite Lebesgue measure in a finite-dimensional space. $\endgroup$
    – Matt R.
    Commented Nov 27, 2018 at 15:49
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    $\begingroup$ Then I'm confused how this is a norm. If $V$ has measure, say, 3, then no function could have norm greater than 3 (taking $s \to 0$). So it can't be homogeneous. Am I missing something? $\endgroup$
    – Nate Eldredge
    Commented Nov 27, 2018 at 16:22
  • $\begingroup$ I guess the authors (who are they anyway?) are talking about the space $S$, also known as $L_0(\mu)$, equipped with the topology of convergence in measure, which can be generated by the metric $d(f,g)=\inf\{\varepsilon>0: \mu\{|f-g|\ge\varepsilon\}\le\varepsilon\}$. Reference to this topology is often made by saying ``in measure'', e.g., one can say that the unit ball of $L_1(\mu)$ is compact in this topology by saying it's compact in measure. $\endgroup$
    – Dirk Werner
    Commented Nov 27, 2018 at 22:45
  • $\begingroup$ In my previous comment I meant to say closed, not compact. There are subspaces of $L_1$ with a unit ball that is compact in measure; see G. Godefroy, N. Kalton, D. Li, J. Reine Angew. Math. 471, 43-75 (1996). $\endgroup$
    – Dirk Werner
    Commented Nov 28, 2018 at 6:13

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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) (Russion version online). 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).

Precise reference: Definition of equivalent metric for convergence in measure, §III.2 of Dunford & Schwartz, Linear Operators, Part I: General Theory (1958). For a theorem actually characterizing compactness in measure see Thm.IV.11.1 in Dunford & Schwartz.

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