Advanced Analysis, Notes 12: Banach spaces (application: existence of Haar measure)

This post is dedicated to my number one follower for her fourteenth birthday,which I spoiled…

Let G be a compact abelian group. By this we mean that G is at once both an abelian group and a compact Hausdorff topological space, and that the group operations are continuous, meaning that g \mapsto g^{-1} is continuous on G and (g,h) \mapsto g+h is continuous as a map from G \times G to G. It is known that there exists a regular Borel measure \mu on G, called the Haar measure, which is non-negative, satisfies \mu(G) = 1, and is translation invariant:

\forall g \in G . \mu(g+ E) = \mu(E) ,

for every Borel set E \subseteq G. In fact, the Haar measure is known to exist in greater generality (G does not have to be commutative and if one allows \mu to be infinite then G can also be merely locally compact). The Haar measure is an indispensable tool in representation theory and in ergodic theory. In this post we will use the weak* compactness of the unit ball of the dual to give a slick proof of the existence of the Haar measure in the abelian compact case.

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