Classification problem: admissible rule is a Bayes rule for some prior $\pi$

Question

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$.

Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in [0,\infty]$, $i\in\{1,2\}$, I want to show that any admissible classification rule is a Bayes classification rule for some prior $\pi$.

Any help doing this would be very much appreciated.