antisigmaalgebra

Antisigmaalgebra is a term found in discussions of dualities in set theory and measure theory, often presented as a hypothetical counterpart to the sigma-algebra. In its conventional formulation, an antisigmaalgebra is defined on a fixed universal set X as a collection A of subsets of X that contains the empty set and X, and is closed under taking complements with respect to X and under countable intersections.

However, a simple consequence of these closure properties shows that an antisigmaalgebra is automatically a sigma-algebra. Because for any countable family {E_n} in A, the union ∪E_n equals the complement of the countable intersection of the complements, i.e., ∪E_n = (∩(E_n)^c)^c, and since E_n^c ∈ A and A is closed under countable intersections, the union ∪E_n belongs to A as well. Therefore, antisigmaalgebras do not yield a distinct mathematical object from sigma-algebras under standard definitions.

Historically or pedagogically, the term is sometimes used to illustrate dualities or to discuss the boundaries between algebraic set systems and measure-theoretic structures. In practice, the concept reinforces that any antisigmaalgebra coincides with a sigma-algebra, and the familiar examples (the trivial collection {∅, X}; any sigma-algebra on X; the full power set) are themselves antisigmaalgebras.

In summary, antisigmaalgebra is not a separate classical object in standard mathematics; it aligns with the theory of sigma-algebras, highlighting dual perspectives rather than introducing a new structure. See also sigma-algebra, measure theory, and Boolean algebra of sets.