Field of sets

In mathematics, a (volume ) Algebra is a key concept in measure theory. He referred to a non-empty class of sets that is, association and komplementstabil.

Felix Hausdorff called because of a distant resemblance to the algebraic structure of a body in number theory a lot of algebra "body", in analogy to its name " ring " for a lot of dressing. Under a ring but is today understood in measure theory a special set lattice, also this concept of the body is significantly different from that of a body in the sense of algebra.

Also, the branch of mathematics that deals with the calculation with quantities is called the algebra of sets. Similarly ambiguous is the term algebra, which is used for a branch of mathematics and also for a special algebraic structure. As used herein, the algebra is however closely related to the Boolean algebra, which is another special algebraic structure.

Definition

Be an arbitrary set. A system of subsets of covering an algebra or algebra if the following properties are satisfied:

Examples

• For any quantity of the smallest and the power set is the largest possible amount of algebra.
• Every σ ​​- algebra is an algebra (but not every algebra is a σ - algebra).

Properties

• Lots of algebra over and always contains the empty set, because contains at least one element and thus are well
• The 6- tuple with the set algebra is a Boolean algebra in the sense of lattice theory, where for all (regarding stability / seclusion average). The empty set corresponds to the zero element and the unit element.

• That any finite union and every finite intersection is comprised of elements of the algebra in it, that is valid for all from the association as well as average stability are each followed by inductively:

Equivalent definitions

If a system of subsets of, and if amounts are then due and the following two statements are equivalent:

• And if also

Identifies beyond the symmetric difference of and so are due and as well as equivalent:

• Is a lot of algebra.
• Is a set lattice and:.
• Is a Boolean algebra.
• Is a ring and quantity.
• Is a lot of half- algebra and:.
• Is a unitary ring in the sense of algebra with addition multiplication and one.
• Is a Boolean ring.
• The scalar is a unitary algebra for the purposes of the ring on the body.
• And we have:.
• And: and.
• And: and.

Related structures

• The Mengenalgebren are exactly the set rings that contain the basic amount. Summarizing quantity rings of a ring within the meaning of the ring with the symmetric difference and the addition and the multiplication on the average, then the Mengenalgebren just the unitary rings ( i.e., one element) of this figure.
• If an algebra is even with respect to the countable union of closed infinite number of its elements, we obtain a σ - (volume ) algebra.