Motivation of Haar Measure

Question

What is the motivation of Haar measure? Where does a measure of abstract topological groups play a role? And why would one come to the conclusion to introduce a measure of these abstract topological groups?

Answer 1

The comments provided very good theoretical insights to the question, but I have the feeling that the author of the post is looking for a more "practical" approach. My answer is certainly not addressing the most important aspects of Haar measure, but it shows an example how it can sometimes be used as a proof technique.

Assume first that $G$ is a finite group. Then every action of $G$ (compatible with the additive structure) on a finite real vector space, or a $G$-invariant convex subset of it, has a fixed point. (Cf. a finite group of congruences of a convex subset of a Euclidean space $\mathbb{R}^n$ has a fixed point that serves as some kind of "center" of the subset.) Namely, pick any vector $v$, and "average out": $\frac{1}{|G|}\sum\limits_{g\in G} g\cdot v$ is a fixed point. Now if $G$ is not finite but compact, a similar idea works. Instead of averaging out, we integrate with respect to the Haar measure $\mu$:

$$\int\limits_{G} g\cdot v \,d\mu$$

This makes sense, since if you focus on a given coordinate, then $g\cdot v$ yields a function $G\rightarrow \mathbb{R}$, so there is no formal obstruction of integration. (OK, maybe the function is not integrable, but often it is.)

More generally, you can obtain a $G$-invariant object this way, preserving some properties of the original object. Check out this really nice application of the idea, which is one of the variants of the so-called Weyl trick (see Theorem 4.1.): https://sites.math.washington.edu/~morrow/336_17/papers17/thomas.pdf

It is used to show that under some conditions, representations can be decomposed into a sum of irreducible representations (see Theorem 4.2.). Of course you can keep saying "OK, but why are representations useful", but at a point, you will lose your audience...

Answer 2

Even if one does not find the abstraction and generality appealing in itself, such general existence-and-uniqueness results are reassuring: even if we anticipate being able to exhibit an explicit or formulaic invariant measure on a tangible topological group of interest, it is comforting to know _in_advance_, _with_certainty_, that there exists such... and that up to scalar multiples it is unique. In practice, in explicit situations, it is indeed often possible to construct an invariant measure... but far less clear that there are no alternatives. (Further, there is the potential confusion between alternative constructions and alternative outcomes...)

The essential uniqueness also means that we can often manipulate Haar measures and measures on quotients $G/H$ without reference to formulaic or construction details, but, rather, just using properties which uniquely characterize them. For example, for unimodular $G$ and $H$, there is a unique measure $d\dot{g}$ on $G/H$ such that, for $f\in C^o_c(G)$ $$ \int_G f(g)\; dg \;=\; \int_{G/H}\Big( \int_H f(\dot{g}h)\;dh\Big)\;d\dot g $$ with Haar measures $dg$ and $dh$... This also applies when $H$ is a discrete subgroup of $G$, and, significantly, shows that we would not need to determine a fundamental domain for the action of $H$ on $G$ to have this integration formula.