Bloom filtering is an important technique for space efficient storage of a conservative approximation of a set S. The set stored may have up to some specified number of “false positive” members, but all elements of S are included. In this paper we consider lossy dictionaries that are also allowed to have “false negatives”. The aim is to maximize the weight of included keys within a given space constraint. This relaxation allows a very fast and simple data structure making almost optimal use of memory. Being more time efficient than Bloom filters, we believe our data structure to be well suited for replacing Bloom filters in some applications. Also, the fact that our data structure supports information associated to keys paves the way for new uses, as illustrated by an application in lossy image compression.