Math Problems Vault

Problem 22.19

Let $\mu$ be a Borel probability measure on a compact group $\mathcal{G}$ . Show that $\mu$ is Haar measure if and only if

\int _{\mathcal{G}} f \circ \varphi _{g} d \mu = \int _{\mathcal{G}} f d \mu \text{ for all } g \in \mathcal{G}, f \in C(\mathcal{G})

where $\varphi_{g}\left(g^{\prime }\right) = g \cdot g^{\prime }$ for all $g^{\prime } \in \mathcal{G}$ .

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Problem 22.20

Let $\mu$ be Haar measure on a compact group $\mathcal{G}$ . Show that $\mu \times \mu$ is Haar measure on $\mathcal{G} \times \mathcal{G}$ .

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Problem 22.21

Let $\mathcal{G}$ be a compact group whose topology is given by a metric. Show that there is a $\mathcal{G}$ invariant metric.

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Problem 22.22

Let $\mu$ be Haar measure on a compact group $\mathcal{G}$ . If $\mathcal{G}$ has infinitely many members, show that $\mu (\left\{ g\right\} ) = 0$ for each $g \in \mathcal{G}$ . If $\mathcal{G}$ is finite, explicitly describe $\mu$ .

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Problem 22.23

Show that if $\mu$ is Haar measure on a compact group, then $\mu (\mathcal{O}) > 0$ for every open subset $\mathcal{O}$ of $\mathcal{G}$ .

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Problem 22.24

Let $S^{1} = \left\{ z = e^{i \theta } \mid \theta \in \mathbb {R}\right\}$ be the circle with the group operation of complex multiplication and the topology it inherits from the Euclidean plane.

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(i)

Show that $S^{1}$ is a topological group.

(ii)

Define $\Lambda = \left\{ (\alpha , \beta ) \mid \alpha , \beta \in \mathbb {R}, 0 < \beta -\alpha < 2 \pi \right\}$ . For $\lambda = (\alpha , \beta ) \in \Lambda$ , define $I_{\alpha } = \left\{ e^{i \theta } \mid \alpha < \theta < \beta \right\}$ . Show that every proper open subset of $S^{1}$ is the countable disjoint union of sets of the form $I_{\lambda }, \lambda \in \Lambda$ .

(iii)

For $\lambda = (\alpha , \beta ) \in \Lambda$ , define $\mu \left(I_{\alpha }\right) = (\beta -\alpha ) / 2 \pi$ . Define $\mu \left(S^{1}\right) = 1$ . Use part (ii) to extend $\mu$ to set-function defined on the topology $\mathcal{T}$ of $S^{1}$ . Then verify that, by Proposition 9 from the preceding chapter, $\mu$ may be extended to a Borel measure $\mu$ on $\mathcal{B}\left(S^{1}\right)$ . (Proposition 9 of Chapter 21: Let $(X, \mathcal{T})$ be a topological space and $\mu : \mathcal{T} \rightarrow [0, \infty ]$ a premeasure. Assume that for each open set $O$ for which $\mu (O) < \infty$ , $\mu (O) = \sup \left\{ \mu (U) \mid U \text{ open and } U \subseteq O\right\}$ . Then every open set is $\mu^{*}$ -measurable and the measure $\mu^{*}: \mathcal{B}(X) \rightarrow [0, \infty ]$ is an extension of $\mu$ .)

(iv)

Show that the measure defined in part (ii) is Haar measure on $S^{1}$ .

(v)

The torus $T^{n}$ is the topological group consisting of the Cartesian product of $n$ copies of $S^{1}$ with the product topology and group structure. What is Haar measure on $T^{n}$ ?

Problem 22.25

Let $\mu$ be a Borel measure on a topological group $\mathcal{G}$ . For a Borel set $E$ , define $\mu^{\prime }(E) = \mu \left(E^{-1}\right)$ , where $E^{-1} = \left\{ g^{-1} \mid g \in E\right\}$ . Show that $\mu^{\prime }$ also is a Borel measure. Moreover, show that $\mu$ is left-invariant if and only if $\mu^{\prime }$ is right-invariant.

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