Regular conditional probability: Difference between revisions

Revision as of 20:45, 1 May 2013

Revision as of 11:52, 24 June 2013

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Thus if ''X'' is distributed uniformly on <math>[0,1],</math> it is truly meaningless to condition a probability on "<math>X=3/2</math>".

==Regularity versus completeness==

{| class=wikitable

! [[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 (mathematics)|Pathological]] cases.

| No pathological cases.

|-

| <math>\lambda(\mathbb Q \cap [0,1])=0.</math>

| <math>\mu(\mathbb Q \cap [0,1])</math> is undefined.

|-

| '''Probability is [[Sigma additivity|<math>\sigma</math>-additive]]'''

| '''except for sets with [[isolated point]]s.'''

|}

Note: In this article we use the [[Fraktur (script)|Fraktur]] <math>\mathfrak P</math> (whose shape is somewhat reminiscent of <math>\mathfrak B</math> 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 [[Mutually exclusive|incompatible]] in a [[separable space]].

==See also==