# A-operation

operation ${\mathcal A}$

A set-theoretical operation, discovered by P.S. Aleksandrov  (see also , ). Let $\{ E _ {n _ {1} \dots n _ {k} } \}$ be a system of sets indexed by all finite sequences of natural numbers. The set

$$P = \cup _ {n _ {1} \dots n _ {k} , . . } \cap _ { k=1 } ^ \infty E _ {n _ {1} {} \dots n _ {k} } ,$$

where the union is over all infinite sequences of natural numbers, is called the result of the ${\mathcal A}$- operation applied to the system $\{ E _ {n _ {1} \dots n _ {k} } \}$.

The use of the ${\mathcal A}$- operation for the system of intervals of the number line gives sets (called ${\mathcal A}$- sets in honour of Aleksandrov) which need not be Borel sets (see ${\mathcal A}$- set; Descriptive set theory). The ${\mathcal A}$- operation is stronger than the operation of countable union and countable intersection, and is idempotent. With respect to ${\mathcal A}$- operations, the Baire property (of subsets of an arbitrary topological space) and the property of being Lebesgue measurable are invariant.

How to Cite This Entry:
A-operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-operation&oldid=50301
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article