Sigma algebra, also known as sigma field or sigma algebra, is a concept in measure theory and probability theory. It is a collection of subsets of a given set that satisfies certain properties. Formally, let X be a set. A σ-algebra on X, denoted by Σ or σ-algebra, is a collection of subsets of X that satisfies the following three properties: 1. The empty set (∅) belongs to Σ. 2. If A is in Σ, then its complement (X - A) is also in Σ. 3. If {A_n} is a countable sequence of sets in Σ, then their union (∪A_n) is also in Σ. In simpler terms, a σ-algebra is closed under taking complements and countable unions. It provides a framework for defining measures and probabilities on the subsets of X. Sigma algebras are essential in probability theory because they allow us to define measurable spaces and probability measures on these spaces. They provide a mathematical structure that enables us to rigorously assign probabilities to events and analyze their properties. Moreover, sigma algebras help us define random variables as measurable functions between measurable spaces. They allow us to work with sets of outcomes rather than individual outcomes, making it easier to study the behavior of random phenomena. Overall, sigma algebras play a fundamental role in measure theory and probability theory by providing the necessary structure for defining measures and probabilities on sets.