dhilbert83 (dhilbert83) wrote in geometry_qa,

Question on Haar Measure/Integration

Ok well you guys said this is for analysis, so don't yell at me since this post is not even remotely geometric :p

If G is a locally compact group and K subse G is compact, and
lim n --> inf int_K f_n(x) dx = 0, then does lim n --> inf int_{K^-1} f_n(x^-1) dx = 0? (f_n subset L^p(G) for some p >= 1) I actually ultimately need this question asked for integration being Bochner integration, but is this true even for Lebesgue integrals? I don't think it is, since the modular function will screw things up even if it's a 'mild' function.
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Are you functions positive? If not, I guess it is possible to make an example of a function f for which the first integral is zero and the second isn't, so f,f,f... will be a counter-example. If they are, I think your claim follows from the fact that the modular function, being continous, is bound on the compact set K.
yes it's easy if they are positive. In general no they are not positive. In the work I'm doing they are actually Banach space valued functions (ie B is a banach space, and f_n : G --> B for each n), so negative and positive have no meaning, which makes it all the more fun :p

Anyway you are probably right. Are there any concrete examples of groups with nontrivial modular function? Maybe some matrix groups?
Let a+b=n, where a,b,n are positive integers. Let P=Pa,bbe the parabolic subgroup of GLn(R) corresponding to the partition (a,b), i.e., the set of matrices of the
             A C
             0 B

The modular function for P is given by   |det A|a |det B|b.

This follows, for example, from a direct computation of the left and right Haar measures, which are equal to
  |det A|-bdAdBdC    and  |det B|-adAdBdC