A useful $σ$-algebra for counter-examples in probability and statistics

Rok, strany: 1996, 55 - 59

Let $Ω$ denote some uncountable set and $A$ the $σ$-algebra, resp. algebra generated by $\{ω\}$, $ω\inΩ\setminus\{ω₀\}$, $ω₀\inΩ$ fixed. Based on $A$ it is shown that

1. in general the Lebesgue decomposition for transition measures does not yield transition measures for the corresponding components,
2. completeness of some sub-$σ$-algebra for a family of distributions is not preserved for the corresponding class of elementary conditional distributions without the additional assumption of sufficiency,
3. there exists an infinite algebra admitting only $σ$-additive probabilities,
4. unique extension of measures beyond complete $σ$-algebras is possible,
5. monogenicity of regular Borel measure does not imply completion regularity,
6. separation of points does not imply measurability of singletons.

ISO 690:

Plachky, D. 1996. A useful $σ$-algebra for counter-examples in probability and statistics. In Tatra Mountains Mathematical Publications, vol. 8, no.2, pp. 55-59. 1210-3195.

APA:

Plachky, D. (1996). A useful $σ$-algebra for counter-examples in probability and statistics. Tatra Mountains Mathematical Publications, 8(2), 55-59. 1210-3195.