4
votes
1answer
180 views

When does convolution preserve the `size' of a function?

For a positive function $f$ and positive measures $\mu, \nu$, does $$\mu\ast f\leq \nu\ast f \Rightarrow \|\mu\|\leq \|\nu\|?$$ More details: Let $G$ be a locally compact group, $C(G)$ be the space ...
4
votes
1answer
153 views

Can Haar measure fail to be bi-invariant without conjugation shrinking a set?

(This is a slightly reformatted and clarified version of my question from math.SE, since I believe the answer there is wrong and its poster has not responded to my comment in over two weeks.) Let ...
1
vote
1answer
254 views

Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral? In general, are there mean value theorems ...
0
votes
1answer
373 views

Positive function with zero Haar integral

If $C$ is a compact (semi-)topological (semi-)group, are there nonzero positive functions having zero Haar integral? In other words: is the Hermitian product associated to the Haar integral ...
2
votes
1answer
816 views

Haar measure of a subgroup

What is the connection between the normalized Haar measure of a compact group and the normalized Haar measure of one of its compact subgroups? I am trying to solve the following problem: Given $G$ ...
21
votes
3answers
934 views

In an inductive family of groups, does the probability that a particular word is satisfied converge?

We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...