Existence of a A measurable function

Let $A$ be sigma algebra having subsets of $R$ only. We define a function from subset of $A$ to $R$ is said to be $A$ measurable iff every Borel set is pulled back to elements of $A$.

Is there a sigma algebra $B$ and a function $f$ such that $f$ is $B$-measurable but there exist some element of $B$ such that inverse image of that doesn't belong to $B$.

(without using existence of non-measurable set Because if non measurable set exist then there exist Lebesgue measurable function $f$ and a measurable set $B$ such that inverse image of $B$ under $f$ is not Lebesgue measurable)