# “Compactness in Measure” in Function Spaces

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.

• 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? – Nate Eldredge Nov 27 at 15:36
• 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. – Matt R. Nov 27 at 15:49
• 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? – Nate Eldredge Nov 27 at 16:22
• 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. – Dirk Werner Nov 27 at 22:45
• 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). – Dirk Werner Nov 28 at 6:13

## 1 Answer

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).