Finite unions are not enough because many natural events in probability involve infinitely many outcomes. A sigma-algebra must be closed under countable unions to ensure that events formed by collecting infinitely many individual outcomes remain measurable.
Conditions: The sample space may be countably infinite.; Events of interest may require combining infinitely many outcomes.
Finite unions are not enough because many natural events in probability involve infinitely many outcomes. A sigma-algebra must be closed under countable unions to ensure that events formed by collecting infinitely many individual outcomes remain measurable.
Conditions: The sample space may be countably infinite.; Events of interest may require combining infinitely many outcomes.