Strukturtheorie Der Wahrscheinlichkeitsfelder Und -Räume
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LEBESGUE hat am Anfang dieses Jahrhunderts den Begriff des Ma es eingeführt, der eine Erweiterung des älteren Begriffes des Inhalts war. So schuf er die Möglichkeit, den Definitionsbereich des Inhaltes auf mehr Teilmengen der Zahlengeraden, allgemeiner eines euklidischen Raumes, zu erweitern. Es entstand dann die Frage, ob jede Teilmenge me bar ist. VITALI entdeckte 1905 die Existenz von nicht me baren Teilmengen der Zahlengeraden. Später, als man abstrakte Ma e auf Mengenkörpern bzw. Booleringen (Booleschen Algebren) einführte, erhob sich, neben dem Problem der Erweiterung des Definitionsbereiches des abstrakten Ma es, auch die Frage, ob man auf einem beliebigen Mengenkörper bzw. Boolering ein nicht triviales Ma , und zwar mit bestimmten Eigenschaf- ten erklären kann. Dies ist im allgemeinen nicht möglich; hier geht wesentlich die Struktur des Mengenkörpers bzw. des Booleringes in das Problem ein. Diese Frage ist für die Wahrscheinlichkeitstheorie von besonderer Bedeutung, weil man in dies r Theorie die Zufallsereignisse mit abstrakten Mengen oder mit Elementen eines Booleringes darstellt und die Wahrscheinlichkeit als ein normiertes und in gewissen Fällen strikt positives Ma betrachtet. Über die Strukturtheorie der Wahr- scheinlichkeitsfelder und -räume sind deshalb in den letzten J ahrzehn- ten viele Untersuchungen angestellt worden. In dem vorliegenden Be- richt haben wir versucht, das Wichtige davon zusammenzustellen. Ein gro er Teil des Berichtes wird den verschiedenen Arten der Unabhän- gigkeit und dem Begriff des cartesischen Produktes gewidmet, Begriffe, die hauptsächlich aus Problemen der Wahrscheinlichkeitstheorie ent- standen sind und eine gro e Rolle bei der Charakterisierung der Struktur der Wahrscheinlichkeitsfelder spielen.
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