We consider a ``length-biased'' shift-dependent information measure, 
related to the differential entropy in which higher weight is assigned to 
large values of observed random variables. This allows us to introduce the 
notions of ``weighted residual entropy'' and ``weighted past entropy'', 
that are suitable to describe dynamic information of random lifetimes, 
in analogy with the entropies of residual and past lifetimes introduced 
in \cite{Eb96} and \cite{DiLo2002}, respectively. 
The obtained results include their behaviors under monotonic transformations.