Relating two measures

Question

Let $f$ be a positive ($f> 0$) Lebesgue integrable function. Then we define a new measure $\nu$ as follows: $$ \nu(E) = \int_{E} f \, dx. $$ Is it true that $$ m(E) = \int_{E} \frac{1}{f} \,d\nu. $$ I am suspeccting this because $d\nu = f \, dx$ so we can write as $dx = f^{-1} \, d\nu$. If this is not correct then how can we "invert" and write $m$ in terms of $\nu$.