What does bimeasurable mean? Is an invertible transformation bimeasurable?

Question

What exactly is the meaning of a bimeasurable transformation? I did not find a very clear answer to that. As far as I see it means that Borelsets are maped to Borelsets.

So an invertible transformation is bimeasurable?

Answer 1

A function $f \colon X \to Y$ between measurable spaces is called bimeasurable if:

1. The set $\ f^{-1}(B) \subset X$ is measurable whenever $B \subset Y$ is measurable.

2. The set $\ f(A) \subset Y$ is measurable whenever $A \subset X$ is measurable.

Note that an invertible measurable function need not be bimeasurable. For example let $X$ be $\mathbb R$ endowed with the Borel $\sigma$-algebra and $Y$ be $\mathbb R$ endowed with the $\sigma$-algebra of all countable sets. Then the identity function from $X$ to $Y$ is measurable (since all countable sets are Borel) but not bimeasurable (Not all Borel sets are countable).