Why is countable additivity important for probability in infinite sample spaces?

In probability theory, countable additivity (or the axiom of countable additivity) is a fundamental property that defines a probability measure. It states that for any sequence of disjoint events $A_1,A_2,\dots$, the probability of their union is the sum of their individual probabilities: $P(A_1\cup A_2\cup \dots )=P(A_1)+P(A_2)+\dots$.1

Its importance in infinite sample spaces stems from several key factors:

• Mathematical Consistency: While finite additivity suffices for finite sample spaces, infinite spaces require a more robust framework to handle limits and infinite series. Countable additivity allows mathematicians to rigorously define probabilities for complex events that cannot be expressed as finite unions.1
• Handling Continuous Outcomes: In experiments with a continuum of outcomes (such as selecting a number from an interval), individual outcomes often have a probability of 0. In these cases, it is impossible to compute the probability of an event by simply "adding" individual outcomes. Countable additivity provides the necessary structure to assign probabilities to sets of outcomes, such as intervals, using Lebesgue measure.12
• Universal Acceptance: Although some theoretical alternatives exist using only finite additivity, countable additivity is virtually universally accepted because it supports the development of powerful, consistent models for probability theory.1