pointwise convergence but not L1 convergence

$\ln (2+N)\int_{\{f_n >N\}} f_n \leq \int_{\{f_n >N\}} f_n \ln (2+f_n)\leq M$ so $\int_{\{f_n >N\}} f_n <\epsilon$ for all $n$ whenver $\frac M {\ln (2+N)} <\epsilon$. This proves that $(f_n)$ is uniformly integrable. (according the the definition in K L Chung's book, for example). Since $f_n \to f$ a.e. it follows that $f_n \to f$ in $L^{1}$.

Edit: To use your definition of uniform integrabilty do the following: Let $\epsilon >0$. Let $\delta=\frac {\epsilon} N$. If $m(A) <\delta$ then $\int_A f_n = \int_{A \cap (f_n >M)} f_n+\int _{A \cap (f_n \leq N)} f_n \leq \int_{(f_n >M)}f_n+Nm(A) <2\epsilon$.