Note on invariant and almost invariant measurable sets

Rok, strany: 2000, 21 - 29

Let $Scr A_j$ stand for $σ$-algebras of subsets of non-empty sets $Ω_j$ and $G_j$ for groups of transformations together with a $σ$-algebra $Scr B(G_j)$ of subsets of $G_j$ admitting some left-invariant probability measure such that the mapping $(g_j,ω_j) o g_j(ω_j)$, $g_jin G_j$, $ω_jinΩ_j$, is $(Scr B(G_j)otimesScr A_j,Scr A_j)$-measurable and the mappings $g_j o gcirc g_j$, $g_jin G_j$, $gin G_j$ fixed, are $(Scr B(G_j),Scr B(G_j))$-measurable, $j=1,2$. Then it is shown that $Scr B(G₁ × G₂, Scr A₁otimesScr A₂) =Scr B(G₁,Scr A₁)otimesScr B(G₂,Scr A₂)$ for compact groups $G_j$ with $Scr L(G_j)$ as the Borel $σ$-algebra of $G_j$, $j=1,2$, and $Scr B(G₁× G₂, Scr A₁otimesScr A₂, Q₁otimes Q₂)=Scr B(G₁,Scr A₁,Q₁)otimesScr B(G₂,Scr A₂,Q₂)$ $[Q₁otimes Q₂]$ is valid. Here $Scr B(G_j,Scr A_j)$ respectively $Scr B(G_j,Scr A_j,Q_j)$ denotes the sub-$σ$-algebra of $Scr A_j$ consisting of all $G_j$-invariant respectively $Q_j$-almost $G_j$-invariant sets $A_jinScr A_j$, i.e., $A_j=g_j(A_j)$, $g_jin G_j$, respectively, $Q_j (A_jΔ g_j(A_j))=0$, $g_jin G_j$, holds true, where $Q_j$ is some $G_j$-invariant probability measure on $Scr A_j$, $j=1,2$. Moreover, in the almost invariant case one might replace countable additivity of the invariant probability measures on $Scr B(G_j)$, $j=1,2$, by finite additivity.

ISO 690:

Plachky, D. 2000. Note on invariant and almost invariant measurable sets. In Tatra Mountains Mathematical Publications, vol. 19, no.1, pp. 21-29. 1210-3195.

APA:

Plachky, D. (2000). Note on invariant and almost invariant measurable sets. Tatra Mountains Mathematical Publications, 19(1), 21-29. 1210-3195.