Questions tagged [haar-measure]

Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.

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Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
Thierry Zell's user avatar
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50 votes
4 answers
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When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
Bugs Bunny's user avatar
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7 votes
4 answers
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Measure on real Grassmannians

OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
Thierry Zell's user avatar
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10 votes
2 answers
3k views

Integration on the space of symmetric matrices

Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its ...
Appliqué's user avatar
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19 votes
2 answers
1k views

Haar measures in Solovay's model

Haar measure is a measure on locally compact abelian groups which is invariants to translations. For example, the Lebesgue measure on the reals is such measure. It can be shown without the use of the ...
Asaf Karagila's user avatar
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19 votes
9 answers
6k views

Haar measure on a quotient, References for

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it (thanks to some comments by Ben Linowitz). Right from the very beginning, Weil ...
Anweshi's user avatar
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12 votes
3 answers
680 views

If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...
David Gao's user avatar
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7 votes
2 answers
2k views

Haar measure on the Grassmannian space

The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $...
timofei's user avatar
  • 297
6 votes
0 answers
81 views

Are invertible measures strictly dense?

Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
Jan_Ch.'s user avatar
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6 votes
1 answer
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Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...
Ali Taghavi's user avatar
5 votes
1 answer
287 views

If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?

Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
Transcendental's user avatar
5 votes
0 answers
212 views

On generically Haar-null sets in the real line

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
Taras Banakh's user avatar
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4 votes
1 answer
225 views

Haar-null union of dense subsets

Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that (Dense $G_{\delta}$) $X_i$ is a dense ...
MrsHaar's user avatar
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4 votes
0 answers
96 views

Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary. To my surprise, it was never answered. So I am elevating it to MathOverflow. I ...
Michael_1812's user avatar
4 votes
1 answer
139 views

Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$?

Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set $$\{(x_1,\dotsc,x_{...
Meisam Soleimani Malekan's user avatar