Uniqueness of Haar Measures 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?
 A: A friend of mine proposed the following definition. For a given locally compact group $G$ let's consider some Haar measure $\mu$ and the space $L^1(\mu)$ of absolutely integrable functions. This space actually does not depend on the specific choice of the Haar measure as $L^1(\lambda\cdot \mu) = L^1(\mu)$ for any $\lambda > 0$.
We can then consider the subspace
$$L^1_0(\mu) = \left\{ f \in L^1(\mu): \int_G f \cdot d\mu = 0 \right\}$$
which is again independent from the specific choice of $\mu$. Let's now define the "space of volumes for $G$"
$$\mathrm{Vol}(G) = L^1 (\mu) / L^1_0(\mu)$$
It is a one-dimensional real vector space. For any measurable set $K$ with finite measure we can assign it a "volume" in $\mathrm{Vol}(G)$ which is trivially the indicator function $\mathbf{1}_K$ modulo any null-integral function. This construction works even for non abelian locally compact groups. For two of these we have
$$\mathrm{Vol}(G \times H) = \mathrm{Vol}(G) \otimes \mathrm{Vol}(H)$$
(where equality stands for canonical isomorphism). For some LCA group we also have
$$\mathrm{Vol}\left(\hat G\right) = \mathrm{Vol}(G)^*$$
Coupled with the fact these spaces are 1-dimensional, this garantees that
$$\mathrm{Vol}\left(G\right) \otimes \mathrm{Vol}\left(\hat G\right) = \mathbb{R}$$
Any Fourier transform can therefore be appropriately typed $\mathcal{F} f \in L^2\left(\hat G, \mathbb{C} \otimes \mathrm{Vol}(G)\right)$ without any non-canonical choice.
