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Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit volume on $G$, in particular when we want to compute a Fourier transform.

However, I have noticed something interesting in the special case where $G = E$ is a finite-dimensional real vector space. In this case you can define a canonical pseudoscalar-valued Haar measure instead of a scalar-valued one. This way, the transform of $f \in \mathcal{L}_2(E,\mathbb{C})$ is now a function $\mathcal{F}f \in \mathcal{L}_2(E^*,\mathbb{C} \otimes_\mathbb{R} \Lambda^{max}E)$. As $\Lambda^{max}E$ is one-dimensional, $(\mathbb{C} \otimes_\mathbb{R} \Lambda^{max}E) \otimes_\mathbb{R} \Lambda^{max}E^*$ is canonically isomorphic to $\mathbb{C}$ which allows the inverse Fourier transform to seamlessly go back to $\mathcal{L}_2(E,\mathbb{C})$.

Of course $\Lambda^{max}G$ is not defined for a general LCA group $G$. One might consider the exterior algebra of $G$ considered as a $\mathbb{Z}$-module. However, I don't think it is possible to define a good notion of "pseudoscalars" this way as there is no good notion of dimension on modules.

So my question is: does anybody know if such an extension exists?

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    @rschwieb: I've not considered any "notion of dimensions for modules" and to be honest I don't know any (except rank, if it included as a notion of dimension). But I'm not even sure that considering LCA groups as $\mathbb{Z}$-modules is the good approach. What I would like to define is a sort of "space of pseudo-scalars" $PG$ on a LCA group $G$. Something on which the tensor product (or a broader notion) works and that would allow me to write $\mathcal{C}\otimes PG \otimes P\hat{G} = \mathcal{C}$.2012-12-30

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