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Published in:
2020 | OriginalPaper | Chapter
1. Preliminaries
Authors : Nicola Gigli, Enrico Pasqualetto
Published in: Lectures on Nonsmooth Differential Geometry
Publisher: Springer International Publishing
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Abstract
In this chapter we introduce several classic notions that will be needed in the sequel. Namely, in Sect. 1.1 we review the basics of measure theory, with a particular accent on the space \(L^0({\mathbf {\mathfrak {m}}})\) of Borel functions considered up to \({\mathbf {\mathfrak {m}}}\)-almost everywhere equality (see Sect. 1.1.2); in Sect. 1.2 we discuss about continuous, absolutely continuous and geodesic curves on metric spaces; in Sect. 1.3 we collect the most important results about Bochner integration. Some functional analytic tools will be treated in Appendix A.