Question: 1.13. Let {X ~7;; t E [0, I]} be a continuous parameter supennartingale. For each t E [0, 1] and sequence {t11} decreasing to t,

1.13. Let {X" ~7;; t E [0, I]} be a continuous parameter supennartingale.

For each t E [0, 1] and sequence {t11} decreasing to t, {X,,,} converges a.e. and in Ll. For each t E [0, 1] and sequence {t11} increasing to t, {X," } converges a.e.

but not necessarily in L I. [HINT: In the second case consider X'" - / {X, I .Jf;,,,}.]

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