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Does there exist a connection between finiteness of a measure and "compactness" of its symmetries?

I'm dealing with a measure $\mu$ defined on a certain space of distributions. I know that this measure is, in some sense, invariant under Möbius transformations. Since the Möbius group is non-compact, this measure has a "non-compact symmetry group".

By a computation I can show that $\mu$ is an infinite measure, but I've heard of a more general principle that if a measure is "too symmetric", that is, the "symmetry group" is non-compact, then the measure is infinite (or something along these lines). For example the Lebesgue measure on $\mathbb{R}$ is invariant under all translations (which we can parametrize by the real numbers, a non-compact group) and this measure is infinite. On the other hand, the Lebesgue measure on the circle $\mathbb{S}$ is finite and invariant under all rotations, which we can think of as a compact group.

I'm not sure how to exactly formulate my question, but roughly speaking the idea is the following. Let $M(X)$ denote the vector space consisting of all measures on a measurable space $X$. Fix a measure $\mu \in M(X)$. Now let $\mathcal{L}(M(X))$ denote the set of all linear operators on $M(X)$ and define $$ S(\mu) := \{ T \in \mathcal{L}(M(X)) : T \mu = \mu\}\,. $$ Is there a connection between finitiness of $\mu$ and compactness of $S(\mu)$? (I'm omitting a lot of topological details here, but I hope that the idea gets communicated)

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There is a notion of this sort in the harmonic analysis of locally compact abelian groups, which generalizes the Fourier transform. Without delving into too many details, the main ideas are:

  1. If $G$ is a locally compact abelian group, there exists a canonical measure on $G$ called the Haar measure, which is invariant under left translation by the group. (Lebesgue measure, for instance, is the Haar measure on the additive group $\mathbb{R}^n$.)

  2. Since we have a measure on $G$, we can think about $L^p$-spaces with respect to this measure. This also allows us to generalize the notion of convolution to any LCA group.

  3. An LCA group $G$ has what is called its Pontryagin dual, the set $\hat{G}$ of all characters of $G$: that is, all continuous homomorphisms $G\to\mathbb{T}$, where $\mathbb{T}$ is the circle group. This is also an LCA group. The major theorem of Pontryagin duality states that an LCA group $G$ is canonically isomorphic to its bidual $\hat{\hat{G}}$.

  4. The Fourier transform can be generalized to LCA groups: basically, instead of integrating an $L^1(G)$ against $e^{i\xi x}$, you integrate against the characters. This gives a function $\hat{f}$ defined on $\hat{G}$. If $\hat{G}$ is $L^1(\hat{G})$, then there is a Fourier inversion formula that lets you recover $f$ from $\hat{f}$. (This comes out from Pontryagin duality: you need to integrate against the characters of $\hat{G}$, but by Pointryagin duality those end up being the points of $G$.)

  5. Here's the kicker: an LCA group $G$ is compact if and only if its dual $\hat{G}$ is discrete. Conversely, $G$ is discrete if and only its $\hat{G}$ is compact. Finally, if $\mu$ is the Haar measure on $G$, then $\mu$ is finite if and only if $G$ is compact. Therefore finiteness of the Haar measure (which is the measure that is intended to respect the symmetries of the group) reflects in the group via compactness, and in the dual by discreteness.

    What does 5. say about the Fourier transform? This is exemplified by the theory of Fourier series on the circle group $\mathbb{T}$, or if you like a finite interval $[0,2\pi]$ under addition mod $2\pi$. This is a compact group, and the Pontryagin dual in this case is $\mathbb{Z}$. This shows us that Fourier series are just one manifestation of the Fourier transform under the broader umbrella of Pontryagin duality: an $L^1(\mathbb{T})$ is sent under the generalized Fourier transform to a function $\hat{f}:\mathbb{Z}\to\mathbb{C}$ (i.e. its list of Fourier coefficients), and the fact that (sufficiently nice) functions can be recovered by summing against Fourier coefficients is just the statement of the generalized Fourier inversion theorem with Pontryagin duality.

This might not be a complete answer, but hopefully it lends some philosophical validity to the ideas you have mentioned.

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  • $\begingroup$ Thank you for your answer. I've thought about learning basics of Fourier analysis on LCA groups but never really got to it. I never knew that it is related to the Haar measure. Now that I think of it, I might be able to show that my measure is finite using these concepts (there might be some nasty details that I'm missing, but the philosophy is clear). I might accept your answer soon if I don't get more answers. $\endgroup$ – desos Dec 20 '16 at 10:51

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