Stückmenge

Stückmenge is a term used in some German mathematical texts to denote a “piece set” or a finite union of pieces that come from a decomposition of a larger set. It expresses the idea of forming a subset by selecting a finite number of basic components, or blocks, from a given partition or decomposition.

Definition. Let X be a set and P = {P_i} a partition of X into disjoint blocks. A Stückmenge S ⊆ X is any subset that can be written as a finite union of blocks from P, namely S = ⋃_{i∈F} P_i for some finite index set F. In this sense, a Stückmenge is a finite union of atoms or blocks of the partition.

Properties. The class of Stückmengen is closed under finite unions: if S1 and S2 are Stückmengen, their union is also a Stückmenge. If each block P_i is measurable, then every Stückmenge is measurable. When the partition is countable, there are countably many Stückmengen.

Examples. In the real line with the standard partition into unit intervals [n, n+1), a Stückmenge could be [0,1) ∪ [2,3) ∪ [5,6). In a measure-theoretic setting, taking finite unions of atoms in a measure algebra yields Stückmengen.

Variants and usage. Some texts broaden the notion to finite unions of elementary sets from a given decomposition, not necessarily a strict partition. The term is more common in German-language sources; English-language references typically describe the concept in terms of finite unions of blocks, atoms, or elements of a partition.

See also. Teilmenge, Partition, Measurable set, Atom, Finite union.