1
$\begingroup$

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

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

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 $\sigma(L^1,L^\infty)$.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Ok, thank you. Can you give me a reference? $\endgroup$
    – guacho
    Commented Feb 26, 2014 at 16:42
  • $\begingroup$ For instance, the first chapter of the old book by P.-A. Meyer "Probability and potentials" $\endgroup$
    – R W
    Commented Feb 26, 2014 at 16:48

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .