Probability Space - Non-atomic Case

Non-atomic Case

If p(ω) = 0 for all ω∈Ω (in this case, Ω must be uncountable, because otherwise P(Ω)=1 could not be satisfied), then equation (∗) fails: the probability of a set is not the sum over its elements, as summation is only defined for countable amount of elements. This makes the probability space theory much more technical. A formulation stronger than summation, measure theory is applicable. Initially the probabilities are ascribed to some “generator” sets (see the examples). Then a limiting procedure allows assigning probabilities to sets that are limits of sequences of generator sets, or limits of limits, and so on. All these sets are the σ-algebra . For technical details see Carathéodory's extension theorem. Sets belonging to are called measurable. In general they are much more complicated than generator sets, but much better than non-measurable sets.