  We study here weighted polynomial multiple ergodic averages. A sequence of
weights is called universally good if any polynomial multiple ergodic average
with this sequence of weights converges in $L^{2}$. We find a necessary
condition and show that for any bounded measurable function $\phi$ on an
ergodic system, the sequence $\phi(T^{n}x)$ is universally good for almost
every $x$. The linear case was understood by Host and Kra.
