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Actually, I've been talking this over with my boss and there's a more direct way of proving it than what I was doing, although it's very much along the same lines.
Instead of looking at an average of averages, simply examine the average of the sums of group elements and try to apply the lemma on that. It's equivialent to what I was suggesting but skips an uneccesary step.
It seems like every time I find a way there's always a better one...
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