Probability Space - General Case

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:

    That sort of half sigh, which, accompanied by two or three slight nods of the head, is pity’s small change in general society.
    Charles Dickens (1812–1870)

    It was a maxim with Mr. Brass that the habit of paying compliments kept a man’s tongue oiled without any expense; and that, as that useful member ought never to grow rusty or creak in turning on its hinges in the case of a practitioner of the law, in whom it should be always glib and easy, he lost few opportunities of improving himself by the utterance of handsome speeches and eulogistic expressions
    Charles Dickens (1812–1870)