On page 29 in Foundations of Modern Analysis Friedman writes roughly:
$f$ is measurable if the inverse image of any open set in $R^1$ is measurable. When $X$ is the Lebesgue (Borel) space, then we say that $f$ is Lebesgue (Borel) measurable on $X$.
He writes this without even labeling it as a definition, and I struggle to see how this is equivalent to the wiki definition "$f$ is measurable if the pre-image of every measurable set is measurable". Is it enough to check for the open sets to see that it is so for all measurable sets?
I interpret Lebesgue (Borel) measurability then as saying all L (B)-measurable sets is mapped by the inverse to a L (B)-measurable set, right? I'm confused.