Invariant Measures

If $\mathcal{T}_{e}$ is a base for the topology at $e$, show that $\left\{ g \cdot \mathcal{O} \mid \mathcal{O} \in \mathcal{T}_{e}\right\}$ is a base for the topology at $g \in \mathcal{G}$.

Show that $K_{1} \cdot K_{2}$ is compact if $K_{1}$ and $K_{2}$ are compact subsets of $\mathcal{G}$.

Let $\mathcal{O}$ be a neighborhood of $e$. Show that there is also a neighborhood $\mathcal{U}$ of $e$ for which $\mathcal{U} = \mathcal{U}^{-1}$ and $\mathcal{U} \cdot \mathcal{U} \subseteq \mathcal{O}$.

Show that the closure $\bar{H}$ of a subgroup $H$ is a subgroup of $\mathcal{G}$.

Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be topological groups and $h: \mathcal{G}_{1} \rightarrow \mathcal{G}_{2}$ a group homomorphism. Show that $h$ is continuous if and only if it is continuous at the identity element of $\mathcal{G}_{1}$.

Use the Banach Contraction Principle to prove Lemma 1.

Lemma 1: Let $E$ be a Banach space and the operator $C \in \mathcal{L}(E)$ have $\left\| C\right\| < 1$. Then $\operatorname {Id} - C$ is invertible and $\left\| (\operatorname {Id} - C)^{-1}\right\| \leq (1 - \left\| C\right\| )^{-1}$.

Use the completeness of $\mathcal{L}(E)$ to show that if $C \in \mathcal{L}(E)$ and $\left\| C\right\| < 1$, then $\sum_{k = 0}^{\infty } C^{k}$ converges in $\mathcal{L}(E)$.

Show that the set of $n \times n$ invertible real matrices with determinant 1 is a topological group if the group operation is matrix multiplication and the topology is entrywise continuity. This topological group is called the special linear group and denoted by $\operatorname {SL}(n, \mathbb {R})$.

Let $H$ be a Hilbert space. Show that an operator in $\mathrm{GL}(H)$ preserves the norm if and only if it preserves the inner-product.

Consider $\mathbb {R}^{n}$ with the Euclidean inner-product and norm. Characterize those $n \times n$ matrices that represent orthogonal operators with respect to an orthonormal basis.

Show that $\operatorname {GL}(E)$ is open in $\mathcal{L}(E)$.

Show that the set of operators in $\mathrm{GL}(E)$ comprising operators that are linear compact perturbations of the identity is a subgroup of $\operatorname {GL}(E)$. It is denoted by $\operatorname {GLc}(E)$.

Show that the adjoint of a representation also is a representation.

Show that a probability functional has norm 1.

Let $E$ be a reflexive Banach space and $K^{*}$ a convex subset of $E^{*}$ that is closed with respect to the metric induced by the norm. Show that $K^{*}$ is weak-* closed. On the other hand, show that if $E$ is not reflexive, then the image of the closed unit ball of $E$ under the natural embedding of $E$ in $\left(E^{*}\right)^{*} = E^{* *}$ is a subset of $E^{* *}$ that is convex, closed and bounded with respect to the metric induced by the norm but is not weak-* closed.

Let $\mathcal{G}$ be a compact group, $E$ a reflexive Banach space, and $\pi : \mathcal{G} \rightarrow \mathrm{GL}(E)$ a representation. Suppose that for each $x \in E$, the mapping $g \mapsto \pi (g) x$ is continuous. Assume that there is a non-empty strongly closed, bounded, convex subset $K$ of $E$ that is invariant with respect to $\pi$. Show that $K$ contains a point that is fixed by $\pi$.

Let $\mathcal{G}$ be a topological group, $E$ be a Banach space, and $\pi : \mathcal{G} \rightarrow \mathrm{GL}(E)$ a representation. For $x \in E$, show that the mapping $g \mapsto \pi (g) x$ is continuous if and only if it is continuous at $e$.

Suppose $\mathcal{G}$ is a topological group, $X$ a topological space, and $\varphi : \mathcal{G} \times X \rightarrow X$ a mapping. For $g \in \mathcal{G}$, define the mapping $\pi (g)$ : $X \rightarrow X$ by $\pi (g) x = \varphi (g, x)$ for all $x \in X$. What properties must $\varphi$ possess in order for $\pi$ to be a representation on $\mathcal{G}$ on $C(X)$ ? What further properties must $\varphi$ possess in order that for each $x \in X$, the mapping $g \mapsto \pi (g) x$ is continuous?

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

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

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

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

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 $\mu$.

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

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.

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.

Does the proof of the Bogoliubov-Krilov Theorem also provide a proof in the case $X$ is compact Hausdorff but not necessarily metrizable?

Let $(X, \mathcal{M}, \mu )$ be a finite measure space and $T: X \rightarrow X$ a measurable transformation. For a measurable function $g$ on $X$, define the measurable function $U_{T}(g)$ by $U_{T}(g)(x) = g(T(x))$. Show that $T$ is measure preserving if and only if for every $1 \leq p < \infty , U_{T}$ maps $L^{p}(X, \mu )$ into itself and is an isometry.

Suppose that $T: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is linear. Establish necessary and sufficient conditions for $T$ to be measure preserving with respect to Lebesgue measure on $\mathbb {R}^{n}$.

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 $\mu$ be Haar measure on this group (see Problem 24). Define $T: S^{1} \rightarrow S^{1}$ by $T(z) = z^{2}$. Show that $T$ preserves $\mu$.

Let $T$ be a measure preserving transformation on a finite measure space ($X, \mathcal{M}, \mu$) and the set $A$ be measurable. Show that for almost all $x \in X$, there are infinitely many natural numbers $n$ for which $T^{n}(x)$ belongs to $A$.