Characterization of discreteness of measures by a property of unique extension

Rok, strany: 1998, 57 - 62

Let $F$ denote the set consisting of all finite subsets of some non-void set $Ω$. It is shown that any set function $Q:F\to [0,1]$ is the restriction of some discrete probability measure on the set $P(Ω)$ consisting of all subsets of $Ω$ if and only if $Q$ can be extended uniquely as a probability charge on $P(Ω)$. Furthermore, it is proved that there exists for some probability measure $P$ on a countably generated $σ$-algebra $A$ of subsets of non-empty set $Ω$ some countable algebra $A'$ generating $A$ such that the restriction of $P$ to $A'$ can be uniquely extended to $A$ among all probability charges as the probability measure $P$ if and only if $P$ coincides with the restriction of some discrete probability measure on $P(Ω)$ to $A$.

ISO 690:

Plachky, D. 1998. Characterization of discreteness of measures by a property of unique extension. In Tatra Mountains Mathematical Publications, vol. 14, no.1, pp. 57-62. 1210-3195.

APA:

Plachky, D. (1998). Characterization of discreteness of measures by a property of unique extension. Tatra Mountains Mathematical Publications, 14(1), 57-62. 1210-3195.