Let f be bounded function, and Fn is decreasing sequence of sigma-fields. Denote Enf=E(f|Fn).
I need inequality of the form P{|En-Einfinity|>a}
Unfortunately, the decay of this sequence can be arbitrarily slow. One can get such a slow decay just by repeating several times one the same sigma-field in the sequence Fn. Say one can start with a sequence F1, F2, .... and generate a new sequence where F1repeats 10 times, F2 repeats 100 times, etc. This way one can get a very slow decay for a fixed function f. One can make more involved constructions without repeating sigma-fields, but also with arbitrarily slow decays.
Thank you for the clear answer. Is there paper (or monograph) where more involved conctructions are considered?
The more involved construction is also not too involved :) Consider [0, 1] with standard Lebesgue measure as the measure space. Let function f be supported on [0,1/2], and let Fn be a sequence of sigma-fields on [0,1/2] with normalized Lebesgue measure. Let the sequence Fn has repetitions, but gives slow decay for approximations in question. Now select an arbitrary decreasing sequence Gn of pairwise different sigma-fields on (1/2, 1] and define new sigma-fields Hn on the whole segment [0,1] as follows: for a fixed n Hn is the smallest sigma-field that contains all elements of Fnand all elements of Gn. Then Hn are already pairwise different, but conditional expectations of f with respect to Hn will be on [0, 1/2] the same as with respect to Fn, and 0 on (1/2,1], so the decay in question will be as slow as for Fn (up to the multiplier 1/2).
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