1.*9. Prove that if {Y n, ,'j6n} is a martingale such that Y n E ,'j6n-l, then for every n, Y n = Y 1 a.e. Deduce from this result that Doob's decomposition (6)

is unique (up to equivalent r.v.'s) under the condition that Zn E 0?;;-1 for every n ::::: 2. If this condition is not imposed, find two different decompositions.