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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.

Requiring the event collection $\mathcal{F}$ to be a sigma-algebra ensures that applying the usual set operations—such as taking complements and countable unions—never produces a set outside of $\mathcal{F}$. This guarantees that if we can measure certain events, we can also measure logically related events like their negations or combinations.

Conditions: $\mathcal{F}$ is intended to be the collection of measurable sets/events in a probability space.

Requiring the event collection $\mathcal{F}$ to be a sigma-algebra ensures that applying the usual set operations—such as taking complements and countable unions—never produces a set outside of $\mathcal{F}$. This guarantees that if we can measure certain events, we can also measure logically related events like their negations or combinations.

Conditions: $\mathcal{F}$ is intended to be the collection of measurable sets/events in a probability space.