TITLE: Family of functions that are bounded in $L^1$ but *NOT* Uniformly Integrable
QUESTION [6 upvotes]: I'm having a difficult time constructing a counter example to this. My intuition (sloppily) is to construct a family of functions {$X_n$} that have Dirac pulses at $n$ and $-n$. Such that $\sup_n \Bbb E[|X_n|\Bbb 1_{X≥n}]=1$
However, I'm not sure if this is correct.

REPLY [5 votes]: The thing which breaks this (and other things like martingale convergence) is the fact that you can have things be very large on small sets. If you have $L^p$ bounds for $p > 1$, this just won't happen. But for $p = 1$, it can! Neal provided a good answer, but allow me to add one that won't require much thinking for you!
$X_n = 2^{n} 1_{[0,2^{-n}]}$
Notice that $||X_n||_{L^1}$ = 1, but no matter how small a set you take, you can find an $n$ large enough where all of your mass will still be there.