  We discuss the problem of the separation and description of the underlying
event (UE) within two existing approaches to UE measurement: the "traditional"
method, widely used at Tevatron, and a recently proposed jet-area/median
method. A simple toy model of UE is developed in order to understand how these
approaches perform. We find that both methods are comparably good for measuring
average properties of the UE but the jet-area/median approach is favorable for
determining fluctuations. We also use the latter method to study the UE from
several existing Monte Carlo generator tunes. We investigate which
characteristics of the underlying event might be useful to measure in order to
improve understanding of its properties and to simulate it well. These include
transverse momentum density per area, intra and inter-event fluctuations and
correlations.
