Given a sample space \(\; \mathcal{X} = \{ w_1, w_2, \ldots, w_n \} \;\) of all possible outcomes, a sigma algebra \(\; \mathcal{F} \;\) is an enumeration of possible events (often all of them) that is self consistent in the sense that we can manipulate its events according to the rules of probability, and stay within the sigma algebra. Each event \(\; A \in \mathcal{F} \;\) is just a set of possible outcomes, \(\; A = \{ w_1, w_2, \ldots, w_n \}, \; w_i \in \mathcal{X} \;\). Sigma algebras are defined because they are useful to define higher constructs, such as probability spaces.
\[ \textrm{Sample space} \;\; \mathcal{X} \;\; \rightarrow \;\; \textrm{Event space}\;\; \mathcal{F} \]For example, consider the sample space of throwing a die, \(\; \mathcal{X} = \{ 1, 2, 3, 4, 5, 6 \} \;\). In this case, the event space would be the set of all subsets of \(\; \mathcal{X} \;\), which includes events such as all throws smaller than 3, \(\; \{ 1, 2 \} \;\), or all even throws, \(\; \{ 2, 4, 6 \} \;\).
Given a set \(\;\mathcal{X}\;\), a \(\sigma\)-algebra \(\;\mathcal{F}\;\), \(\;\mathcal{E}\;\) or \(\; \Sigma \;\) is a non-empty collection of subsets of \(\;\mathcal{X}\;\) such that:
NOTE A sigma algebra is just a subset of the powerset that includes itself, is closed under complement, and is closed under addition. Thus, for every set there is only one powerset, but there are many possible sigma algebras.