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:
“There has always been the same amount of light in the world. The new and missing stars, the comets and eclipses, do not affect the general illumination, for only our glasses appreciate them.”
—Henry David Thoreau (1817–1862)
“The landscape of the northern Sprawl woke confused memories of childhood for Case, dead grass tufting the cracks in a canted slab of freeway concrete. The train began to decelerate ten kilometers from the airport. Case watched the sun rise on the landscape of childhood, on broken slag and the rusting shells of refineries.”
—William Gibson (b. 1948)