Why in topological groups that: $E$ is Borel iff $E^{-1}$ is Borel.

Question

Let $G$ be a topological group and $\mu$ a measure on sigma algebra generated by the topology of $G$. A borel set is every element of that sigma algebra. I have found this statement: $E$ is a Borel set iff $E^{-1}$ is a Borel set.

I have no idea about why this is true. I feel that I should use the fact that inversion is homeomorphism in topological groups but I don't know how to proceed next.

Any idea would be appreciated.

Answer 1

It's just that if $f$ is continuous and $B$ is a Borel set, then $f^{-1}(B)$ is also a Borel set.