If for all (in this case, Ω must be uncountable, because otherwise could not be satisfied), then equation () fails: the probability of a set is not necessarily the sum over the probabilities of its elements, as summation is only defined for countable numbers 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.
If for all (in this case Ω must be uncountable because otherwise could not be satisfied) then equation () fails the probability of a set is not necessarily the sum over the probabilities of its elements as summation is only defined for countable numbers 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