1.8. Let {XIl , .~, n EN} be a submartingale and let ex be a finite optional r.v. satisfying the conditions:

(a) / (IXal) < 00, and (b)

lim r IXI1 I d./j) = O. '1~00 J{a>n}

Then {X aM, '~/\Il; 11 E N ex;} is a submartingale. [HINT: for A E '-~An bound fA (Xu -Xal\ll) d.~ below by interposing XaA11l where n < m.]