measure absolutely continuous with respect to the lebesgue measure for which $f_n \in L^1$ but $\{f_n\}$ does not converge

Question

Let $f_n = \frac{(n^2x^2)^{1/2}-nx+x}{(1+x^4)atan(nx)} + \frac{n}{n+nx^2+nx^3}$

Find (if exists) a measure $\mu$ absolutely continuous with respect to the lebesgue measure for which $f_n \in L^1(\Bbb R_+,\mu)$ for each $n$, but $\{f_n\}$ does not converge in $L^1(\Bbb R_+,\mu)$.

I've no ideas anche the main problem is that I don't know measures different form the lebesgue one, except the counting measure...

Thanks for your help.