  In a recent paper \cite{CHR16}, Chandra, Hu and Rosalsky introduced the
notion of a sequence of random variables being uniformly nonintegrable, and
presented a list of interesting results on this uniform nonintegrability. In
this note, we introduce a weaker definition on uniform nonintegrability (W-UNI
for short) of random variables, present a necessary and sufficient condition
for W-UNI, and give two equivalent characterizations of W-UNI, one of which is
a W-UNI analogue of the celebrated de La Vall\'{e}e Poussin criterion for
uniform integrability. In addition, we give some remarks, one of which gives a
negative answer to the open problem raised in \cite{CHR16}.
