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:
“We will win this war because we are willing to sacrifice 10,000,000 lives. How many lives is the White Man willing to sacrifice?”
—Jerome Cady, U.S. screenwriter, and Lewis Milestone. General Mitsubi (Richard Loo)
“The doctors are all agreed that I am suffering for want of society. Was never a case like it. First, I did not know that I was suffering at all. Secondly, as an Irishman might say, I had thought it was indigestion of the society I got.”
—Henry David Thoreau (1817–1862)