I'm interested in knowing whether there exists any kind of theory for measures on groups without assuming that it's the Haar measure for a locally compact group topology. 



measures on groups without assuming a locally compact group topology
Translastion-invariant, locally finite measure on a topological group. 


As I recall, there is something like this in Hewitt & Ross Vol. I



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> *Hewitt, E.; Ross, K. A.*, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Group representations, Die Grundlehren der mathematischen Wissenschaften. 115. Berlin-Göttingen-Heidelberg: Springer-Verlag. VIII, 519 p. (1963). [ZBL0115.10603](https://zbmath.org/?q=an:0115.10603).
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What the exact conditions were, I do not remember. But the conclusion was that the measure is essentially Haar measure on a locally compact group which is in sense a "completion" of the given group.


