Haar measures are decomposable
$begingroup$
The definition of decomposable measures is as follows: (Here $mathcal{M}$ is a $sigma$-algebra over $X$.)
My question is part c) of the following exercise:
I have managed to prove a) and b). For c) I guess the decomposition $mathcal{F}$ (using the notation of the above definition) should be given as follows: Take a clopen $sigma$-compact subgroup $H$ as in a). Then $H$ is a countable union of Borel sets of finite measure. Now these sets and their left translations constitute a partition of $G$, and this is our $mathcal{F}$.
However, I could not prove (iii) or (iv).
My attempt: I have no idea how to verify (iii). For (iv) it's a corollary of the following proposition (which I think is true but could not prove): Let $X$ be a topological space and $Asubseteq X$. If the intersection of $A$ with every connected component of $X$ is a Borel set, then so is $A$.
Any hints will be appreciated!
real-analysis measure-theory borel-sets haar-measure
asked Jan 23 at 5:27
ColescuColescu
3,20011136
$endgroup$
add a comment |
$begingroup$
The definition of decomposable measures is as follows: (Here $mathcal{M}$ is a $sigma$-algebra over $X$.)
My question is part c) of the following exercise:
I have managed to prove a) and b). For c) I guess the decomposition $mathcal{F}$ (using the notation of the above definition) should be given as follows: Take a clopen $sigma$-compact subgroup $H$ as in a). Then $H$ is a countable union of Borel sets of finite measure. Now these sets and their left translations constitute a partition of $G$, and this is our $mathcal{F}$.
However, I could not prove (iii) or (iv).
My attempt: I have no idea how to verify (iii). For (iv) it's a corollary of the following proposition (which I think is true but could not prove): Let $X$ be a topological space and $Asubseteq X$. If the intersection of $A$ with every connected component of $X$ is a Borel set, then so is $A$.
Any hints will be appreciated!
real-analysis measure-theory borel-sets haar-measure
asked Jan 23 at 5:27
ColescuColescu
3,20011136
$endgroup$
add a comment |
$begingroup$
The definition of decomposable measures is as follows: (Here $mathcal{M}$ is a $sigma$-algebra over $X$.)
My question is part c) of the following exercise:
I have managed to prove a) and b). For c) I guess the decomposition $mathcal{F}$ (using the notation of the above definition) should be given as follows: Take a clopen $sigma$-compact subgroup $H$ as in a). Then $H$ is a countable union of Borel sets of finite measure. Now these sets and their left translations constitute a partition of $G$, and this is our $mathcal{F}$.
However, I could not prove (iii) or (iv).
My attempt: I have no idea how to verify (iii). For (iv) it's a corollary of the following proposition (which I think is true but could not prove): Let $X$ be a topological space and $Asubseteq X$. If the intersection of $A$ with every connected component of $X$ is a Borel set, then so is $A$.
Any hints will be appreciated!
real-analysis measure-theory borel-sets haar-measure
asked Jan 23 at 5:27
ColescuColescu
3,20011136
$endgroup$
The definition of decomposable measures is as follows: (Here $mathcal{M}$ is a $sigma$-algebra over $X$.)
My question is part c) of the following exercise:
I have managed to prove a) and b). For c) I guess the decomposition $mathcal{F}$ (using the notation of the above definition) should be given as follows: Take a clopen $sigma$-compact subgroup $H$ as in a). Then $H$ is a countable union of Borel sets of finite measure. Now these sets and their left translations constitute a partition of $G$, and this is our $mathcal{F}$.
However, I could not prove (iii) or (iv).
My attempt: I have no idea how to verify (iii). For (iv) it's a corollary of the following proposition (which I think is true but could not prove): Let $X$ be a topological space and $Asubseteq X$. If the intersection of $A$ with every connected component of $X$ is a Borel set, then so is $A$.
Any hints will be appreciated!
real-analysis measure-theory borel-sets haar-measure
real-analysis measure-theory borel-sets haar-measure
asked Jan 23 at 5:27
ColescuColescu
3,20011136
asked Jan 23 at 5:27
ColescuColescu
3,20011136
asked Jan 23 at 5:27
ColescuColescu
3,20011136
asked Jan 23 at 5:27
ColescuColescu
3,20011136
asked Jan 23 at 5:27
ColescuColescu
3,20011136
3,20011136
add a comment |
add a comment |
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