Let $f_{n}$ be a family of nonnegative Lebesgue measurable functions on [0,1] with the property that $\sup\int_{0}^{1}f_{n}\ln(2+f_{n})\leq M$ for some constant M. Suppose $f_{n}$ converges pointwise a.e. to a measurable function $f:[0,1]\to \mathbf{R}$. Prove or disprove the claim that $||f_{n}-f||_{1}\to 0$ as $n\to \infty$.
I can prove that the integrals of $f_{n}$ are bounded by 2M and $f$ is integrable. I have tried some easy functions like characteristic functions and the claim seems to be true. I don't know whether there is a counterexample.
$\ln (2+N)\int_{\{f_n >N\}} f_n \leq \int_{\{f_n >N\}} f_n \ln (2+f_n)\leq M$ 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 $\int_A f_n = \int_{A \cap (f_n >M)} f_n+\int _{A \cap (f_n \leq N)} f_n \leq \int_{(f_n >M)}f_n+Nm(A) <2\epsilon$.