My problem is with an alternative proof on Harris Inequality ( if A and B are increasing events then P(A,B) > P(A)P(B) . But I was curious about not proving it using expectation. First I said that it is true for cylinder events ( obvious) but as I went further I came up with the fact that if I want to generalize it using approximation of measure, then the approximation events might not be increasing events. Is it true at all that for some increasing event there is an approximation (with a sequence of measurable sets in the generator of sigma algebra) which all of its sets are increasing? Or we can just prove Harris inequality using expectation ( FKG theorem) ?
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