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.

oblomov_jerusalAugust 13 2005, 16:25:56 UTC 11 years ago

dhilbert83August 13 2005, 16:33:03 UTC 11 years ago

Anyway you are probably right. Are there any concrete examples of groups with nontrivial modular function? Maybe some matrix groups?

bravchickAugust 13 2005, 17:57:54 UTC 11 years ago

a+b=n,wherea,b,nare positive integers. LetP=Pbe the parabolic subgroup of_{a,b}GLcorresponding to the partition_{n}(R)(a,b),i.e., the set of matrices of theform

A C0 B

The modular function for

Pis 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|

^{-b}dAdBdC and |det B|^{-a}dAdBdCrespectively.

oblomov_jerusalAugust 13 2005, 18:02:05 UTC 11 years ago

http://www.google.co.il/url?sa=t&ct=res&cd=5&url=http%3A//www.math.utah.edu/%7Emilici

At p.80 it states that some group is not unimodular.