# Questions tagged [haar-measure]

Use this tag for questions related to the Haar measure, which is an assignment of an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

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### Haar measure and push forward in the subset of squares

Let $K$ be a compact group with Haar probability measure $m$, let $K^2 = \{k^2 : k\in K\}$ and suppose that $m(K^2)>0$. Show that if $m(A\cap K^2) = m(K^2)$ then $m(\{k\in K : k^2\in A\})=1$.
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### Absolute continuity of the push forward of the Haar measure

Let $K$ be a compact group with Haar measure $m$. Let $K^2 = \{k^2 : k\in K\}$ and suppose that $K = K^2 \cup wK^2$ for some $w\in K$. Let $\mu$ be the push-forward of $m$ under the map $k\mapsto k^2$....
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### Integration over the general linear group over $\mathbb Q_p$

Before asking, I apologize if this question is duplicated. It seems a sort of basic stuff and I tried to find the reference, but I couldn't find it(only the statement without the proof appears in the ...
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### $L^1(G)$ is a Banach *-Algebra [duplicate]

Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
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### Computing Integral over the Orthogonal Group $O(d)$

Question: I am seeking methods or references to numerically compute integral over the orthogonal group $O(d)$. The specific context is to compute integrals of the form: $$\int_{O(d)} f(g) dg$$ where ...
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### Understanding definition of Adelic integral and calculate simple example

I'm trying to understand the definition the adelic integral given in Goldfeld and Hundley. It says that: Suppose that $f =\prod_v f_v$ is a factorizable function, that $f_\infty$ is an integrable ...
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### Haar measure on the positive real numbers

Consider the locally compact Hausdorff topological group $\mathbb{R}_{>0}$, I can use Haar theorem to show that there is a Haar measure on my borel sets. However, I am having trouble determining it....
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Here is a problem I am trying to solve. Let $G$ be a compact group acting continuously on a locally compact metrix space $X$. Suppose $\mu$ is an $G$- invariant ergodic Borel probability measure of $... • 87 0 votes 0 answers 62 views ### Convolution of Double Coset Indicator Function in Hecke Algebra of Locally Profinite Group Let$G$be a locally profinite group (i.e. a topological group that is locally compact Hausdorff and totally disconnected or, equivalently, a Hausdorff topological group s.t.$1 \in G$has a ... • 23 1 vote 1 answer 138 views ### Integration of periodic functions on the real line with respect to the Haar measure of 1-dim. torus A function$f:\mathbb R \to \mathbb C$with period 1 can be identitied with a function defined on the 1-dimensional torus$\mathbb T = \mathbb R / \mathbb Z$, the latter being continuous if and only ... 1 vote 0 answers 27 views ### Why is the$\text{weak}^*$-closedness important when considerning convex combinations of invariant Borel probability measures? I got a question while reading page 148 of Katok and Hasselblat. Suppose$\mu$is an$L_{g_0}$-invaraint Borel probability measure for the translation$L_{g_0}(h) = g_0h$on a compact metrizable ... • 1,005 1 vote 0 answers 95 views ### Generating Random Linear Subspace Let$n$and$m$be two integers such that$m\leq n$. Let$G_{n,m}$be the set of all m-dimensional linear subspaces of$\mathbb{R}^n$. Assume we want to generate a subspace of dimension$m$which is ... • 714 1 vote 0 answers 51 views ### Explicit expressions for the integrals of second degree polynomials in the orthogonal group Wikipedia has explicit formulas for first and second degree polynomial integrals in the unitary group using Weingarten functions (https://en.wikipedia.org/wiki/Weingarten_function). Is there any ... 2 votes 1 answer 74 views ### How often is a monic polynomial highly divisible by p? Let$p$be a prime, and let$|\cdot|_p$be the$p$-adic absolute value. Let$f(x) \in \mathbb{Z}[x]$be a monic polynomial with$f(0) = 1$. Question 1 What is the volume$c_n$of the following set $$\{... • 571 2 votes 1 answer 39 views ### Why is \Delta(f)(s,t)= f(st) well-defined? Let G be a locally compact group with Haar measure \lambda. The locally compact group G\times G then carries the Haar measure \lambda\times \lambda. One can then define the map$$\Delta: L^\... • 882 3 votes 0 answers 186 views ### Every Haar measure is a multiple of the counting measure. I was asked to prove the following: Let$G$be a group, equipped with the discrete topology. A Haar measure on$G$is a measure$\mathcal{P}(G) \rightarrow [0, \infty]$such that:$$\mu(K) < \... • 121 5 votes 1 answer 251 views ### Are Haar measures localizable? I'm trying to prove that Haar measures are localizable. we know that Haar measures are decomposable ( see Haar measures are decomposable) in the sense that: A measure space$(X,\mathfrak{M},\mu)$is ... 9 votes 0 answers 322 views ### Haar measures are decomposable In the real analysis book by Folland, section$11.1$exercise$9$have been come that: if$G$is a locally compact topological group with Haar measure$\mu$, then$\mu$is decomposable. A measure ... 2 votes 0 answers 120 views ### If the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero$\mu(\{x\})=0$in folland-real analysis,chapter 11.1, exercise$9$have been come that: if the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero meaning that$\...
Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $\mu$ a Borel measure on $X$. To be sure, continuity of the action ...
We know that semifinite measure space $(X,\mathcal{M},\mu)$ is a measure space that for every measurable set $E\in\mathcal{M}$ with measure $\mu(E)=\infty$, there exist a measurable set $B\subseteq E$...