# Tagged Questions

140 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 ...
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### Question on the Quotient Integral Formula

I have a very concrete question on the proof of the following (see below): Given a 'nice' top. space $X$ and a 'nice' group operation of $G$, say, from the right, on $X$ and a certain measure on $X$, ...
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### How do these two Haar measures on SL(2,R) compare?

By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as ...
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### Is function from topological group to metric space Borel?

Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact metric space and $f:X\rightarrow G$ a continuous bijective function. Suppose there exists $g\in G$ such that if ...
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### 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 ...
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### Difference between spaces of integrable functions w.r.t Lebesgue measure and Borel measure [closed]

Is there a difference between $L^p(\mathbb R,\mathfrak B,\beta)$ and $L^p(\mathbb R,\mathfrak L,\lambda)$ ? Here I denoted by $\lambda$ the Lebesgue measure, defined on the Lebesgue $\sigma$-algebra ...
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### Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices? My understanding for what Haar measure means for $U(1)$ is that it ...
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### Measures and structure on conjugacy classes

Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$  \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} ...
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### Volume of fundamental domain and Haar measure

In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...
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### exotic compact group

Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...
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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 ...
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### Is every probability space a factor space of the Haar Measure on some group?

Let P be an arbitrary probability space. I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$. By a ...
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### Haar Measure Existence/A problem with Borel sets and regularity.

In Paul Halmos' Measure Theory book, section 53, he defines a content on a locally compact Hausdorff space to be a set function, $\lambda$ that is additive, subadditive, monotone, and ...
Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have ...