May Composition of a strongly monic and a measurable functions be unmeasurable?

May Composition of a strongly monic and a measurable functions be
unmeasurable?

I'm interested to know that is there any strongly monic function like g
and a lebesgue measurable set E, such that $g^{-1}(E)$ be not Lebesgue
measurable. I think it should exist but I couldn't construct it. I will be
happy if someone here help me with it.