(A pet peeve of mine is Mathematicians from field X noticing that field Y uses terminology which is very close to that from field X, and assuming there are Mathematical links. This question might be along these lines).
In the literature around Fourier transforms and multipliers on Abelian groups, the term "pseudomeasure" occurs. For example, Section 4.2 of Larsen's book defines a "pseudomeasure" to be bounded linear functional on $A(G)$ the Fourier algebra of $G$. Here $G$ is a locally compact abelian group and so $A(G)$ is identified with $L^1(\widehat G)$ by the Fourier transform, and so pseudomeasures are simply the "Fourier transforms" of elements of $L^\infty(\widehat G)$. I guess the terminology arises because genuine measures on $G$ give rise to functionals on $A(G)$ by integration. (I have to say that my personal intuition is that this view is not wildly helpful.) The terminology has then been copied for non-abelian groups, and generalisations to $L^p$ spaces; see for example Chapter 4 of Derighetti's book.
By contrast, in Section 6.3 of Folland's book a psuedomeasure is a continuous linear functional on $C_{00}(G)$ the space of compactly supported continuous functions with the inductive limit topology. Such objects, to me, do seem to be more closely linked to measures. Their use here is in generating unitary representations, by way of the notion of a positive definite pseudomeasure.
My questions are:
Is there a genuine mathematical link between these two notions, or are they just superficially similar?
I would be interested in the history of why "psuedomeasure" for the dual of $A(G)$: who first used this term, and can one guess at the motivation for it?
