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
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