General Case
If Ω is uncountable, still, it may happen that p(ω) ≠ 0 for some ω; such ω are called atoms. They are an at most countable (maybe, empty) set, whose probability is the sum of probabilities of all atoms. If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case. Otherwise, if the sum of probabilities of all atoms is less than 1 (maybe 0), then the probability space decomposes into a discrete (atomic) part (maybe empty) and a non-atomic part.
Read more about this topic: Probability Space
Famous quotes containing the words general and/or case:
“In democratic ages men rarely sacrifice themselves for another, but they show a general compassion for all the human race. One never sees them inflict pointless suffering, and they are glad to relieve the sorrows of others when they can do so without much trouble to themselves. They are not disinterested, but they are gentle.”
—Alexis de Tocqueville (1805–1859)
“There are a great many of these accusers, and they have been accusing me now for a great many years, and what is more, they approached you at the most impressionable age, when some of you were children or adolescents; and literally won their case by default, because there was no one to defend me.”
—Socrates (469–399 B.C.)