SigmaAlgebra

A σ-algebra on a set $X$ is a collection $\mathcal{F}$ of subsets of $X$ that is closed under countable unions, countable intersections, and complements. This means that:

• If $A$ is in $\mathcal{F}$, then the complement of $A$ is also in $\mathcal{F}$. Here, the complement of $A$ in $X$ is the part of $X$ that, together with $A$, completes $X$, aligning with the notion of mutual completion.
• If $(A_n)$ is a countable sequence of sets in $\mathcal{F}$, then the union and intersection of the $A_n$ are also in $\mathcal{F}$.

These properties ensure that a σ-algebra is a suitable structure for defining measures, such as probability measures, on the sets it contains. Since a σ-algebra is closed under countable unions and intersections, as well as complements, it is particularly suitable for defining measures in a clear and rigorous manner. In probability theory, a σ-algebra represents the collection of events that can be assigned probabilities, and the underlying set $X$ is the sample space consisting of all possible outcomes.