Lower bound on the correlation between monotone families in the average case

A well-known inequality due to Harris and Kleitman [T.E. Harris, A lower bound for the critical probability in a certain percolation process, Math. Proc. Cambridge Philos. Soc. 56 (1960) 13-20; D.J. Kleitman, Families of non-disjoint subsets, J. Combin. Theory 1 (1966) 153-155] states that any two monotone subsets of {0, 1}ⁿ are non-negatively correlated with respect to the uniform measure on {0, 1}ⁿ. In [M. Talagrand, How much are increasing sets positively correlated? Combinatorica 16 (2) (1996) 243-258], Talagrand established a lower bound on the correlation in terms of how much the two sets depend simultaneously on the same coordinates. In this paper we show that when the correlation is averaged over all the pairs A, B ∈ T for any family T of monotone subsets of {0, 1}ⁿ, the lower bound asserted in [M. Talagrand, How much are increasing sets positively correlated? Combinatorica 16 (2) (1996) 243-258] can be improved, and more precise estimates on the average correlation can be given. Furthermore, we generalize our results to the correlation between monotone functions on [0, 1]ⁿ with respect to the Lebesgue measure.