Integral and measures on manifolds

I would like to have a good reference about integrals on immersed manifolds in euclidean space. The setting is the following: let $F:M \rightarrow \mathbb{R}^m$ be an immersion and $\rho: \mathbb{R}^m \rightarrow \mathbb{R}$ be a well-behaved function. Then we can consider the integral $$ \int_M \rho d\mu, $$ where $d\mu$ is the measure defined by the Riemannian metric on $M$ inherited from $\mathbb{R}^m$ by $F$.
Multiple references are also good, together they should include:

• Details about the construction of $d\mu$ (I think it is defined using the Riesz representation theorem).
• Convergence results: if we have convergence $F_n \rightarrow F$ (w.r.t. to some topology), when do the integrals also converge?
• A relation between $d\mu$ and the Hausdorff measure.

Do we sometimes need completeness of $M$ to obtain the desired results?

Solution 1:

I think a good reference for a constructive approach to integrals and measures on immersed manifolds in $\mathbb{R}^N$ could be the book [1] by Ambrosio, Fusco and Pallara. I hesitated to post this answer since I own a copy of the book only from the first days of January, therefore I cannot provide a comprehensive review of it. Also I cannot access the MR review of the book, so I cannot say if it says something useful to answer your question, and on the other hand, the Zbl review focuses on the analysis of the second part of the book, while the part which is of interest to you is the first one, consisting of chapters 1 and 2 (the first 115 pages of the book). However, referring to the comment of Diesirae92, to your observations pertaining it and to the three points emphasized in your question as guidelines for a review, I can put forward the following remarks.

1. The first chapter is an introduction to "abstract" measure theory: I think it is not useless to stress that the term "abstract" in this chapter should be intended according to the Italian tradition of Picone, Tonelli, Fichera, De Giorgi and their schools as "with the widest conceivable applicability". The Carathéodory construction is given (§1.4, p. 23-24) and in the second chapter, dealing with basic geometric measure theory, it is shown how to construct the Hausdorff measure $\mathscr{H}^k$, $k\leq N$ in $\mathbb{R}^N$ by using it (§2.8, pp. 72-79): for the relation between $d\mu$ and this measure see point 3 below. Incidentally, also Riesz's theorem is proved as a consequence of Carathéodory criterion (§1.4, p.25), and the Area and Co-area formula are proved (respectively §2.10, pp. 87-92 and §2.12, 101-108), describing the theory of measure and integration on various measurable subsets of the Euclidean $N$-space.

2. Concerning the convergence of a family of immersions $\{F_n\}$ to an immersion $F$, the very same chapter describe the weak convergence in $L^p$ spaces (§1.2, pp. 15-18) and the weak* convergence of measures (§1.4, p. 26-29). Although the theory of those kind of convergence is not applied explicitly to the convergence of families of immersions, it is used throughout chapter 2. There is also an application to the theory of Young measures, dealing with the problem of calculating the weak* limit of a family of functions $\{f(u_h)\}$ where $f:\mathbb{R}^M \to \mathbb{R}$ is a simple continuous function, and $u_h: E\subset\mathbb{R}^N\to\mathbb{R}^M$, is a family of bounded $L^1$ functions.

3. The equivalence between $d\mu$ and the Hausdorff measure $\mathscr{H}^k$ can be proved by Besicovitch differentiation theorem (§2.4, pp.54-55), according to the remark of Diesirae92: the authors made also an interesting remark on the integration respect to a Hausdorff measure (§2.8, p. 75), stating that the Hausdorff measure is not even $\sigma$-finite if $k<N$.

Other personal notes

• The approach taken in this book is not differential geometric but rather measure theoretic and analytic in spirit. However, even not dealing explicitly with immersions of manifolds, it describes quite sharply the structure of various very general sets in the Euclidean space and integrals/measures defined on them, images of measurable sets under suitably defined low-smoothness mappings.
• As stated in point 1 above, the stile is "abstract" in the Italian sense, i.e. very general but still focused on applicability and application. This means that a reader with a strong calculus background should (and will) be able to understand it.

[1] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000). Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. New York and Oxford: The Clarendon Press/Oxford University Press, New York, pp. xviii+434, ISBN 0-19-850245-1, MR1857292, Zbl 0957.49001.