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Characterizing the metric compactification of \(L_p\) spaces by random measures. (English) Zbl 1451.54005

The metric compactification \(\overline{X}^h\) of a metric space \((X,d)\) is defined as the pointwise closure of the family of \(1\)-Lipschitz functions \(\{h_y:y\in X\}\), where \(h_y(\cdot)=d(\cdot,y)-d(x_0,y)\) and \(x_0\in X\) is a fixed arbitrary point. In the case in which \(X\) is proper and geodesic, the injection \(y\mapsto h_y\) is a continuous embedding and \(\overline{X}^h\) coincides with Gromov’s horofunction compactification of \(X\). For a general metric space \(X\), the map \(y\mapsto h_y\) is only a continuous injection so \(\overline{X}^h\) is a compactification of \(X\) in a weaker sense.
The paper provides a complete characterization of the metric compactification of non-atomic \(L_p\) spaces for \(1\leq p<\infty\), showing that the elements of \(\overline{L_p}^h\) are represented by means of random measures on the compact Polish space \(\overline{\mathbb{R}}^h\). That representation should be compared to the one of \(\overline{\ell_p}^h\) obtained by the author in [Can. Math. Bull. 62, No. 3, 491–507 (2019; Zbl 1435.54009)].
As an application, the paper presents new proofs of the \(L_p\)-mean ergodic theorem for \(1<p<\infty\) and D. E. Alspach’s example of an isometry with no fixed points [Proc. Am. Math. Soc. 82, 423–424 (1981; Zbl 0468.47036)].

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
60G57 Random measures
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References:

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