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This paper considers space-efficient data structures for storing an approximation S' to a set S such that S ⊆ S' and any element not in S belongs to S' with probability at most ∈. The Bloom filter data structure, solving this problem, has found widespread use. Our main result is a new RAM data structure that improves Bloom filters in several ways:• The time for looking up an element in S' is O(1), independent of ∈.• The space usage is within a lower order term of the lower bound.• The data structure uses explicit hash function families.• The data structure supports insertions and deletions on S in amortized expected constant time.The main technical ingredient is a succinct representation of dynamic multisets. We also consider three recent generalizations of Bloom filters.