A sigma-algebra is a mathematical structure used in measure theory, which is a branch of mathematics that deals with the concept of size or measure. It is a collection of sets that satisfies three key properties: it includes the entire space, is closed under complementation, and is closed under countable unions. This means if you have a set in the sigma-algebra, its complement is also in the sigma-algebra, and you can combine countably many sets from the sigma-algebra to form a new set that is also included. Sigma-algebras are essential in defining measures, such as the Lebesgue measure, which helps in understanding concepts like probability and integration. They provide a framework for working with infinite collections of sets, making them crucial in fields like probability theory and statistics. By ensuring that certain operations on sets yield results within the same collection, sigma-algebras help maintain consistency in mathematical analysis.