Regularity Versus Completeness
| Standard probability space | Radon space |
|---|---|
| Lebesgue measure | Borel measure |
| Complete measure | Regular measure |
| Conditional probability | Regular conditional probability |
| Extremely complicated and weak. | Simple and powerful. |
| Pathological cases. | No pathological cases. |
| is undefined. | |
| Probability is -additive | except for sets with isolated points. |
Note: In this article we use the Fraktur (whose shape is somewhat reminiscent of for Borel) to indicate a probability based on a regular measure as opposed to one based on a complete measure. The notions of regularity and completeness are incompatible in a separable space.
Read more about this topic: Regular Conditional Probability
Famous quotes containing the words regularity and/or completeness:
“Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.”
—Charles Sanders Peirce (1839–1914)
“Poetry presents indivisible wholes of human consciousness, modified and ordered by the stringent requirements of form. Prose, aiming at a definite and concrete goal, generally suppresses everything inessential to its purpose; poetry, existing only to exhibit itself as an aesthetic object, aims only at completeness and perfection of form.”
—Richard Harter Fogle, U.S. critic, educator. The Imagery of Keats and Shelley, ch. 1, University of North Carolina Press (1949)