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:
“The general fact is that the most effective way of utilizing human energy is through an organized rivalry, which by specialization and social control is, at the same time, organized co-operation.”
—Charles Horton Cooley (1864–1929)
“The attention of those who frequent the camp-meetings at Eastham is said to be divided between the preaching of the Methodists and the preaching of the billows on the back side of the Cape, for they all stream over here in the course of their stay. I trust that in this case the loudest voice carries it. With what effect may we suppose the ocean to say, “My hearers!” to the multitude on the bank. On that side some John N. Maffit; on this, the Reverend Poluphloisboios Thalassa.”
—Henry David Thoreau (1817–1862)