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I read the article "Komlós Theorem for Unbounded Random Sets" by G. KRUPA (MSN), but I did not understand the difference between:

  • Measurable multifunction integrably bounded,

  • Measurable multifunction integrable

(see p. 239).

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Let $F:\Omega\to 2^X$ be a measurable multifunction with closed values in the separable Banach space $X$. It seems to be implicitly assumed that $F$ jas nonempty values.

$F$ is integrably bounded if the function $\omega\mapsto \sup_{x\in F(\omega)}\|x\|$ is integrable.

$F$ is integrable if the function $\omega\mapsto \inf_{x\in F(\omega)}\|x\|=d(0,F(\omega))$ is integrable.

The point of these definitions might be better understood in terms of measurable selections. By the Kuratowski-Ryll-Nardzewski measurable selection theorem, $F$ admits measurable selections. If $F$ is integrably bounded, all selections will be integrable and dominated by a single integrable function. If $F$ is integrable, at least one selection will be integrable.

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  • $\begingroup$ Let $\mathcal{P}_{c}(X)$ be the family of all closed subsets of $X$. We know that the space of all measurable multifunctions $F:\Omega\to \mathcal{P}_{c}(X)$ integrably bounded noted by $\mathcal{L}^1_{\mathcal{P}_{c}(X)}$. But what is the notation of the space of all the measurable multifunctions integrable? $\endgroup$
    – kaka Hae
    Commented May 10, 2020 at 4:19
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    $\begingroup$ I don't think the author introduces special notation for integrable multifunctions. $\endgroup$
    – Michael Greinecker
    Commented May 10, 2020 at 6:07
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    $\begingroup$ The multifunction with constant value $X$. $\endgroup$
    – Michael Greinecker
    Commented May 10, 2020 at 15:48
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    $\begingroup$ I think "Infinite Dimensional Analysis" by Aliprantis and Border is a great source for material on multifunctions (correspondences in the book). $\endgroup$
    – Michael Greinecker
    Commented May 11, 2020 at 18:43
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    $\begingroup$ Measurability of a multifunction is generally not defined in terms of a $\sigma$-algebra on $2^X$. In special cases, something like that is possible though. If $X$ is a separable metrizable space and $\mathcal{K}$ the family of nonempty compact subsets of $X$ and if $f:\Omega\to 2^X$ has values in $\mathcal{K}$, then measurbility of $f$ corresponds to measurability with respect to the Hausdorff metric topology on $\mathcal{K}$. This is Theorem 18.10 in the mentioned book. $\endgroup$
    – Michael Greinecker
    Commented May 12, 2020 at 7:39

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