Show that Theorem 24.6 is enough to prove the Radon-Nikodm theorem (Theorem 25.2 ) for a countably

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Show that Theorem 24.6 is enough to prove the Radon-Nikodým theorem (Theorem 25.2 ) for a countably generated \(\mathscr{A}\), i.e. \(\mathscr{A}=\sigma\left(\left\{A_{n}ight\}_{n \in \mathbb{N}}ight)\).

[ set \(\mathscr{A}_{n}:=\sigma\left(A_{1}, A_{2}, \ldots, A_{n}ight)\) and observe that the atoms of \(\mathscr{A}_{n}\) are of the form \(C_{1} \cap\) \(\cdots \cap C_{n}\), where \(C_{i} \in\left\{A_{i}, A_{i}^{c}ight\}, 1 \leqslant i \leqslant n\).]

Data from theorem 25.2

Theorem 25.2 (Radon-Nikodm) Let u, v be two measures on the measurable space (X, A). If  is o-finite, then

Data from theorem 24.6

Theorem 24.6 (convergence of UI submartingales) Let (un)neNo be a submartin- gale on the o-finite filtered

For the last estimate we use that on the set {u>Lw} n {ut

(ii)  (iii). Because of uniform integrability we have for some  >0 and a suitable w E L (A) [lunds= ==

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