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Asgar Jamneshan and I have just uploaded to the arXiv our paper “Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration“. This paper arose from our longer-term project to systematically develop “uncountable” ergodic theory – ergodic theory in which the groups acting are not required to be countable, the probability spaces one acts on are not required to be standard Borel, or Polish, and the compact groups that arise in the structural theory (e.g., the theory of group extensions) are not required to be separable. One of the motivations of doing this is to allow ergodic theory results to be applied to ultraproducts of finite dynamical systems, which can then hopefully be transferred to establish combinatorial results with good uniformity properties. An instance of this is the uncountable Mackey-Zimmer theorem, discussed in this companion blog post.

In the course of this project, we ran into the obstacle that many foundational results, such as the Riesz representation theorem, often require one or more of these countability hypotheses when encountered in textbooks. Other technical issues also arise in the uncountable setting, such as the need to distinguish the Borel {\sigma}-algebra from the (two different types of) Baire {\sigma}-algebra. As such we needed to spend some time reviewing and synthesizing the known literature on some foundational results of “uncountable” measure theory, which led to this paper. As such, most of the results of this paper are already in the literature, either explicitly or implicitly, in one form or another (with perhaps the exception of the canonical disintegration, which we discuss below); we view the main contribution of this paper as presenting the results in a coherent and unified fashion. In particular we found that the language of category theory was invaluable in clarifying and organizing all the different results. In subsequent work we (and some other authors) will use the results in this paper for various applications in uncountable ergodic theory.

The foundational results covered in this paper can be divided into a number of subtopics (Gelfand duality, Baire {\sigma}-algebras and Riesz representation, canonical models, and canonical disintegration), which we discuss further below the fold.

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A key theme in real analysis is that of studying general functions {f: X \rightarrow {\bf R}} or {f: X \rightarrow {\bf C}} by first approximating them by “simpler” or “nicer” functions. But the precise class of “simple” or “nice” functions may vary from context to context. In measure theory, for instance, it is common to approximate measurable functions by indicator functions or simple functions. But in other parts of analysis, it is often more convenient to approximate rough functions by continuous or smooth functions (perhaps with compact support, or some other decay condition), or by functions in some algebraic class, such as the class of polynomials or trigonometric polynomials.

In order to approximate rough functions by more continuous ones, one of course needs tools that can generate continuous functions with some specified behaviour. The two basic tools for this are Urysohn’s lemma, which approximates indicator functions by continuous functions, and the Tietze extension theorem, which extends continuous functions on a subdomain to continuous functions on a larger domain. An important consequence of these theorems is the Riesz representation theorem for linear functionals on the space of compactly supported continuous functions, which describes such functionals in terms of Radon measures.

Sometimes, approximation by continuous functions is not enough; one must approximate continuous functions in turn by an even smoother class of functions. A useful tool in this regard is the Stone-Weierstrass theorem, that generalises the classical Weierstrass approximation theorem to more general algebras of functions.

As an application of this theory (and of many of the results accumulated in previous lecture notes), we will present (in an optional section) the commutative Gelfand-Neimark theorem classifying all commutative unital {C^*}-algebras.

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In the next few lectures, we will be studying four major classes of function spaces. In decreasing order of generality, these classes are the topological vector spaces, the normed vector spaces, the Banach spaces, and the Hilbert spaces. In order to motivate the discussion of the more general classes of spaces, we will first focus on the most special class – that of (real and complex) Hilbert spaces. These spaces can be viewed as generalisations of (real and complex) Euclidean spaces such as {\Bbb R}^n and {\Bbb C}^n to infinite-dimensional settings, and indeed much of one’s Euclidean geometry intuition concerning lengths, angles, orthogonality, subspaces, etc. will transfer readily to arbitrary Hilbert spaces; in contrast, this intuition is not always accurate in the more general vector spaces mentioned above. In addition to Euclidean spaces, another fundamental example of Hilbert spaces comes from the Lebesgue spaces L^2(X,{\mathcal X},\mu) of a measure space (X,{\mathcal X},\mu). (There are of course many other Hilbert spaces of importance in complex analysis, harmonic analysis, and PDE, such as Hardy spaces {\mathcal H}^2, Sobolev spaces H^s = W^{s,2}, and the space HS of Hilbert-Schmidt operators, but we will not discuss those spaces much in this course.  Complex Hilbert spaces also play a fundamental role in the foundations of quantum mechanics, being the natural space to hold all the possible states of a quantum system (possibly after projectivising the Hilbert space), but we will not discuss this subject here.)

Hilbert spaces are the natural abstract framework in which to study two important (and closely related) concepts: orthogonality and unitarity, allowing us to generalise familiar concepts and facts from Euclidean geometry such as the Cartesian coordinate system, rotations and reflections, and the Pythagorean theorem to Hilbert spaces. (For instance, the Fourier transform is a unitary transformation and can thus be viewed as a kind of generalised rotation.) Furthermore, the Hodge duality on Euclidean spaces has a partial analogue for Hilbert spaces, namely the Riesz representation theorem for Hilbert spaces, which makes the theory of duality and adjoints for Hilbert spaces especially simple (when compared with the more subtle theory of duality for, say, Banach spaces). Much later (next quarter, in fact), we will see that this duality allows us to extend the spectral theorem for self-adjoint matrices to that of self-adjoint operators on a Hilbert space.

These notes are only the most basic introduction to the theory of Hilbert spaces.  In particular, the theory of linear transformations between two Hilbert spaces, which is perhaps the most important aspect of the subject, is not covered much at all here (but I hope to discuss it further in future lectures.)

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