On Haar Measures Carson Teitler, nct2119 December 11th, 2020 Contents 1 Introduction and Motivation 1 2 Measures 2 2.1 Borel σ-Algebra . .2 2.2 Regular Measure . .3 2.3 Extension of Premeasures . .4 3 Riesz Markov Theorem 5 3.1 Linear Functionals . .5 3.2 Dual Space . .5 3.3 Riesz-Markov Theorem . .5 4 Topological Group 9 4.1 Translations in Topological Groups . 10 4.2 Representations of Topological Groups . 12 4.3 Kakutani's Fixed Point Theorem . 13 5 Existence of Unique Haar Measures for Compact Groups 15 6 Examples and Properties of the Haar Measure 16 6.1 Haar Measures of Singleton Sets in Topological Groups . 16 6.2 Haar Measure of Measurable Subgroups . 17 6.3 Open Subsets have Positive Measure . 17 6.4 On the Complex Circle . 18 6.5 On the Quaternions Sphere . 19 7 Applications of the Existence of Haar Measures 19 7.1 There are no compact infinitely countable topological groups . 19 7.2 Group invariant metrics . 20 7.3 Measures induced by transitive group action . 22 8 Bibliography 25 1 Introduction and Motivation The concept of a measure in mathematics is one that in many context codifies an intuitive idea mathematically. On the real number line, for example, the Lebesgue measure extends the idea of length or size in English to a large family of sets, called the Borel sets, which are combinations of intersections, complements, and unions applied to open and closed intervals. In particular, the Lebesgue measure matches the intuitive sense of Length for any unit interval, assigning the intervals ra; bs and pa; bq both |b ´ a| length. One difficulty with measure, though, is that it requires a lot of artifice to construct in a manner that acts in accordance with many properties 1 one would want in a mathematical understanding of size. Some of those difficulties will be discussed in the first few sections of this paper, but suffice it to say for now that even when there is a somewhat natural expectation of what a measure should look like on a space, it often takes some pretty complex and idiosyncratic mathematics to write down what the measure actually is and does. This difficulty is what makes the Haar measure such a powerful tool. It equips us with a measure on a fairly large class of mathematical spaces, namely compact topological groups.1 Even more importantly, Haar measures are translation-invariant, meaning that if we have the set S and then define the sets gS as the elements of S left multiplied by g in our compact topological group, and similarly for right multiplication and Sg then we find that each of these three sets have the same measures under the Haar measure. There is an intrinsic motivation for this, which is that groups often represent symmetries of different systems, and therefore heuristically we would likely want measures to be symmetric under the action of the group as well, but this justification for importance may still be too abstracted and at first glance, invariance may seem mildly arcane or unimportant; however consider what this actually means on specific spaces. If we return to the understanding of the Lebesgue measure on R as length, then R under addition is a topological group. What invariance tells us here on an interval is that mpr0; 1srq “ mpr0; 1sq, which is to say mprr; r ` 1sq “ mpr0; 1sq, or essentially if we slide the interval up or down the number line without stretching it, it keeps the same length. In this context, it is clear why our mathematical definition of a length function or a size function should naturally want to be translation invariant. These measures, or size functions, do not exist solely for their own sake, either. There are several external appeals for the construction of the Haar measure on any compact topological group. The construction of measures for these spaces allows us automatically to integrate any continuous function on any compact group. This follows almost immediately from the construction of such a measure. Additionally, Haar measures are the unique invariant Borel probability measure on the topology. Therefore if we wish to understand the group as a probability space with an invariant measure (e.g. studying random events invariant under rotations), something one might naturally desire when studying symmetry, the Haar measure is the unique such measure. Additionally, the Haar measure is a relative nice measure that, in conjunction with a Radon-Nikodym derivative, can express a integration over a large number of probability spaces over the group. The existence of these measures can inform us about topological groups more generally as well. For example we will prove that no countable topological group is compact (see Section 7.1). We can also use these measures to generate measures on large classes of spaces using group actions, and we can generate group invariant metrics for compact topological metric groups. 2 Measures The goal of this portion of the paper is to acquire the necessary machinery to construct Haar measures. This means that sometimes the more computational aspects of certain topics will be covered, such as the constructions of certain functions, sets or other objects but some of their properties will be left unproven with a citation to a text with a proof. In this section we will introduce some definitions and theorems that aid in the construction of a Haar measure, but will assert several properties which are important but nevertheless relate more directly to studies of measures generally than the study of Haar measures. This is because many of the measure theoretic concepts in the beginning of this chapter should generally be familiar to a reader of this text, or any student who had taken an introductory real analysis course; however, this section is included here for clarity of notation and definitions. 2.1 Borel σ-Algebra If the goal of this section is to establish the set-functions known as measures, we must first define what measures act upon. We naturally want the idea of size to be additive and to have some idea of subtraction as well, which would indicate that we would want unions, intersections, and complements of measurable sets to be measurable. This intuition matches up nicely with the domain of measures, namely σ-algebra. Defintion 2.1. [5] Let X be a set. Then A is a σ-algebra if for any countable index set I of some sequence Ai P A, we have that 1This concept can be extended more broadly to locally compact topological groups, but the construction of such measures is outside the scope of this paper, and generally even more involved than the compact case. 2 1. iPI Ai P A 2. ŞiPI Ai P A c 3. ŤAi P A In general, if F is a family of sets in X, then σpFq denotes the σ-algebra generated by F, which is the smallest σ-Algebra that contains all elements of F. The existence of such a σ-algebra is not trivial, and is outside the scope of this paper, but is often proved using the π ´ λ theorem. With our domain now defined, let us proceed to provide an exact definition of these set functions which roughly correspond to size, called measures Defintion 2.2. [5] A Measure is a function from a σ-algebra A with µ : A Ñ r0; 8s such that µpHq “ 0 and for some countable index set I and a disjoint sequence of sets Ai P A, we have µ Ai “ µpAiq ˜iPI ¸ iPI ¤ ¸ We identify a topological space with its measure as a measure space and write pX; F; µq where X is the space, µ is the measure, and F is the σ-algebra. Measures have a particularly helpful use, which is in the integration of functions. In order to integrate continuous functions to R, it helps to have a measure over the closed sets of a space, since then f ´1ptfpxquq is closed and can be measured (A reader is expected to know the basics of measure theoretic integration. If not, refer to [5]). This of course, given the condition on complements, means that the open sets of the space will also be measurable. Correspondingly, we will define Defintion 2.3. The Borel σ-Algebra of a topological space X with topology T is the σ-algebra BpXq :“ σpT q. Correspondingly, a Borel measure on a space X is a measure on the Borel σ-algebra of X, µ : BpXq Ñ R Y t8u such that if S Ă X is compact, then µpSq ă 8. Note that every open and closed set is thus contained in the Borel σ-algebra, because if U is open then U P T Ă σpT q and if U is closed then U c P T , and therefore by closure under complements we have U “ pU cqc P T . Building a Borel σ-Algebra on a generic space can be somewhat complicated, but it does exist and is the σ-algebra generated by the topology, i.e. the collection of open sets, on a given space. This lines up with our desire for a measure for the purpose of integration. 2.2 Regular Measure Defintion 2.4. [3, p. 455] Let F be a σ-algebra on a topological space X. Then a measure µ is deemed regular if the following two conditions are met: • (Outer Regularity) If U P F, then µpUq “ inftµpOq | U P F;U Ă O is openu • (Inner Regularity) If U Ă X is open and U P F then µpUq “ suptµpKq | K P F;K Ă U is compactu A regular Borel measure is called a Radon measure.