To what extent has the Haar measure been generalized? It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly wider class of groups that has such a measure? What about weakening of these measures? What about hypergroups?
Thank you. HTTT.
 A: For hyperfinite groups - ultraproducts $G$ of sequences of finite groups $G_n$ - one has the Loeb probability measure, which is analogous to Haar measure in that it is a bi-invariant probability measure.  The catch though is that the Loeb measure is not measurable with respect to a Borel sigma algebra (indeed, there is no natural topology to place on a hyperfinite group), but instead on the Loeb sigma algebra generated by applying the Caratheodory construction to the Boolean algebra of internal sets (the ultraproduct of subsets $E_n$ of $G_n$, which have measure equal to the (standard part of the) ultralimit of $|E_n|/|G_n|$).
One way to think of this is that while a hyperfinite group $G$ is not a topological space, it is a sigma-topological space - it has a collection of "open" sets (countable union of internal sets) which obey a weakened version of the topology axioms in which one has closure only under countable unions rather than arbitrary unions.  Loeb measure is then the analogue of Haar measure with respect to this sigma-topology.  I discuss this a bit in my paper with Bergelson.
One can also generalise from the hyperfinite case to ultraproducts of compact groups, or even locally compact groups if one normalises things properly.
A: For general locally compact hypergroups, the existence of a Haar measure is still an open problem as far as I know.  It has been answered affirmatively for Abelian, compact, or discrete hypergroups and those arising as coset spaces from locally compact groups.  
A: For a Hausdorff topological group, it is equivalent to admit a (quasi-)leftinvariant Radon measure and being locally compact.
Reference: http://www.ams.org/journals/proc/1972-035-02/S0002-9939-1972-0318777-5/S0002-9939-1972-0318777-5.pdf
I don't know about Hypergroups.
It seems that quotient of topological groups give Hypergroups: Note that $G/H$ admits in general only a $G$-invariant measure if the modular functions of $G$ and $H$ coincide on $H$. I am not sure if that helps in your context.
