# On an existence of a quasi-finite left- invariant Borel measure in a non-locally compact Polish group

Let $(G,B(G))$ be a Polish group. A Borel set $A \subset G$ is called Haar null if there is a Borel probability measure $\mu$ in $G$ such that $\mu(g(A))=0$ for each $g \in G$.

A Borel measure $\lambda$ in $G$ is called left invariant if $\lambda(g(X))=\lambda(X)$ for each $X \in B(G)$ and $g \in G$.

A Borel measure $\nu$ in $G$ is called quasi-finite if there is a compact set $F\subset G$ such that $0<\nu(F)<+\infty$.

Question. Let $(G,B(G))$ be a non-locally compact Polish group and $A$ be Haar null set. Does there exist a quasi-finite left-invariant Borel measure $\mu$ in $G$ such that $\mu(A)=0$?

-
One standard reference on Haar null sets is Solecki's article. It is behind a paywall, so I can't tell you whether your precise question is discussed in that article. Maybe you can find something in the articles available on his homepage. – Martin Jan 2 '13 at 11:24
Does there always exist a quasi-finite left-invariant Borel measure for a non-locally compact Polish group? – Ilya Jan 2 '13 at 12:22

## 1 Answer

An answer on Ilya question is yes. See [John C. Oxtoby, Invariant Measures in Groups Which are not Locally Compact, Transactions of the American Mathematical Society, Vol. 60, No. 2 (Sep., 1946), pp.215-237] Theorem 3, p. 220.

-