On diffeologies for power sets and measures. (English) Zbl 1543.58018
This paper aims to provide a suitable framework for differentiation on the power set of a space, in which the flavor of rigor and adeptness is comforting enough to enable to think about a general theory adapted to applied frameworks. For this purpose, the authors work in the framework of diffeology [P. Iglesias-Zemmour, Diffeology. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1269.53003)]. Diffeological spaces form a category for differential calculus and differential geometry, an interesting subcategory of which Frölicher spaces [A. Frölicher and A. Kriegl, Linear spaces and differentiation theory. Chichester (UK) etc.: Wiley (1988; Zbl 0657.46034); A. Kriegl and P. W. Michor, The convenient setting of global analysis. Providence, RI: American Mathematical Society (1997; Zbl 0889.58001)] form.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- (1)
- §2 recalls basics on diffeologies and Frölicher spaces.
- (2)
- §3–§5 provide and study in detail samples of diffeologies on power sets with desirable properties. There are three refinements of the power set diffeology in [P. Iglesias-Zemmour, Diffeology. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1269.53003)] \[ \text{Strong diffeology}\hookrightarrow\text{Union diffeology}\hookrightarrow \text{Weak diffeology} \] Having diffeological structures on power sets, one can easily treat smooth set-valued maps as usual smooth maps in the diffeological setting.
- (3)
- §6 describes so-called projectable diffeologies on power sets, developed in the same spirit as the \(Diff\)-diffeology [J.-P. Magnot, Nonlinearity 33, No. 12, 6835–6867 (2020; Zbl 1454.39031)], for which we have \[ \text{Locally projectable diffeology}\hookrightarrow\text{Union diffeology} \]
- (4)
- §7.1 establishes the existence of diffeologies on the power set \(\mathfrak{B}\left( X\right) \) for which the Boolean operations are smooth. §7.2 and §7.3 analyze how diffeologies on a Borel algebra may encode differentiabiligy or smoothness of measures in it. §7.4 defines a diffeology on the space of measures.
- (5)
- §8 and §9 are concerned with a possible framework for set-valued maps, showing how Hadamard derivatives of functionals on shape analysis fit with a directional derivative of the globally projectible diffeology.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 58-XX | Global analysis, analysis on manifolds |
| 28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |
| 47H04 | Set-valued operators |
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