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Haar measure on locally sigma-compact metric groups

$G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar measure on a locally compact Hausdorff group, I strength the condition Hausdorff property(modified as a metric space), and weaken the locally compact condition(modified as a locally sigma-compact/locally countable-compact group).

My question is, dose there exist a (left) Haar measure on a locally sigma-compact metric group? Dose there exist a (left) Haar measure on a locally countable-compact metric group?

Or what conditions can be weaken to make a group still have a Haar measure?

A topological space is countably compact if every countable open cover has a finite subcover.

A topological space is said to be σ-compact if it is the union of countably many compact subsets.

A subset $A$ of $X$ is said to be σ-compact if it is the union of countably many compact subsets in $X$.

$X$ is a locally sigma-compact sapce, if all points has a neibourhood which is sigma-compact, that is to say for any $x\in X$, there exist a neibourhood $U$ of $x$, and $U$ is a sigma-compact subset of $X$.

$X$ is a locally countable-compact sapce, if all points has a neibourhood which is countable compact, that is to say for any $x\in X$, there exist a neibourhood $U$ of $x$, and $U$ is a countable compact subset of $X$.

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It seems the following.

I am not an expert in this question, but I think you may encounter some problems defining Haar measure on such a group.

One of ways to define a (Haar?) measure $\mu$ on a non-locally compact group $G$ may be to consider a completion $\hat G$ of the group $G$. If the group $\hat G$ is locally compact and has a Haar measure $\hat\mu$, we may try to define a (Haar?) measure $\mu$ on the group $G$ by putting $\mu(A)=\hat\mu(\operatorname{cl}_{\hat G}(A))$ for each closed set $A\subset G$.

For instance, by such a way we may define an invariant measure $\mu$ on a topological group $\Bbb Q$ of rationals. By the measure $\mu$ is not $\sigma$-additive, because the group $G$ is a countable union of singletons (that is, one-point sets), each of which has measure zero.

Update. Prof. Banakh said that good measures on non-locally compact topological groups are not known. But there are subadditive measures and capacity measures, like Solecki submeasures. There are his papers, which may be useful for you:

T. Banakh, Extremal densities and measures on groups and $G$-spaces and their combinatorial applications.

T. Banakh, The Solecki submeasures and densities on groups.

T. Banakh, I. Protasov, S. Slobodianiuk, Densities, submeasures and partitions of groups, Algebra Discr. Math. 17:2, (2014) 193–221.

T.Banakh, I.V.Protasov, S.Slobodianiuk, Syndetic submeasures and partitions of $G$-spaces and groups, IJAC 23:7 (2013), 1611–1623.

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  • $\begingroup$ $\mu(A)=\hat\mu(\operatorname{cl}_{\hat G}(A))$, in general, $\hat\mu$ has some properties? $\endgroup$ – David Chan May 22 '15 at 12:08
  • $\begingroup$ @DavidChan It is hard for me to tell something wise about that, so I forward your question to my professor Taras Banakh. If I remember it right, he with Nadya Lyaskovska once used a similar construction. $\endgroup$ – Alex Ravsky May 22 '15 at 13:57
  • $\begingroup$ @DavidChan I obtained an answer from Prof. Banakh, see an update of the answer. $\endgroup$ – Alex Ravsky May 24 '15 at 15:18

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