I think that if a sequence of L^1 functions have the integral 
$$
\int f\_n \log (f\_n)dx
$$
uniformly bounded, then there is a subsequence that converges strongly in $L^1$. 


The questions are:


1) Is this correct?


2) Can you give me a good reference for this topic?


Thank you in advance



L logL space and compactness
1) No. Take the sequence of functions $f\_n$ on the unit interval defined in the following way: $f\_n(x)=1$ (resp., 2) if the $n$-th digit in the dyadic decomposition of $x$ is 1 (resp., 0).


2) What is true is following. Your condition (or rather, uniform boundedness of $\int |f\_n|\log^+|f\_n|$ implies that the sequence $f\_n$ is **uniformly integrable**, which by **Dunford-Pettis theorem** is equivalent to relative compactness of the sequence $f\_n$ in the **weak topology** $\sigma(L^1,L^\infty)$. 


