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A subset of a topological space $X$ is called nowhere dense in $X$ if the interior of its closure is empty.

A subset of a topological space $X$ is called the first category (or meagre) in $X$ if it is a union of countably many nowhere dense subsets.

A subset of a topological space $X$ is called the second category (or nonmeagre) in $X$ if it is not of first category in X.

$G$ is a topological group, $A$ and $B$ are the subsets of $G$, we denote $AB$=$\{ab:a \in A, b\in B \}$, and $A^{-1}=\{a^{-1}: a \in A\}$.

My question is:

Let $G$ be a locally compact (Hausdorff) topological group, and $A$ and $B$ are two Borel subsets(generated by open subsets) of the second category in $G$, then $AB^{-1}$ must contain a non-void open subset.

Is there any reference book about this proposition?

Thanks in advance.

David Chan
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1 Answers1

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I recall that a subset $B$ of a topological space $X$ has the Baire Property in $X$ if $B$ contains a $G_\delta$-subset $C$ of $X$ such that $B\setminus C$ is meager in $X$. By [Kech, 8.22] each Borel subset of a space $X$ has the Baire Property in $X$. By Banach-Kuratowski-Pettis Theorem (see [Kel, p.279] or [Kech 9.9]), for any subset $B$ of a topological group the set $BB^{-1}$ is a neighborhood of the identity $e$ provided $B$ is non-meager and has the Baire Property in the group.

References

[Kech] A. Kechris. Classical Descriptive Set Theory, – Springer, 1995.

[Kel] J. Kelley. General Topology. -- M.: Nauka, 1968 (in Russian).

Alex Ravsky
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