Prove that if real function $f$ is measurable on $R$ then for every borel set $B$ preimage $f^{-1}(B)$ is measurable.
Proof in my textbook goes like this:
We know from previous problem that if $f$ is measurable then for every open set $U$ $f^{-1}(U)$ is measurable. Set X of subsets of $R$ such that preimage is measurable is sigma algebra is itself sigma algebra (easy to check). So X contain all open sets. Because family of borel sets is the smallest sigma algebra which contain all open sets then family of borel sets is contained in X.
My doubt is what if family of borel sets contain some set that is not in X however is still smaller? Then family of borel sets is NOT contained in X.