I am reading Chapter 2.2 of Percolation by Grimmett on the FKG Inequality and I am having a hard time visualizing the meaning of this inequality when I interpret it the statement in terms of integrals.
In particular a corollary of the FKG Inequality is stated as:
If A and B are increasing events then $P_p(A \cap B) \geq P_p(A)P_p(B)$
Where $P_p$ here is the product measure on the $d$ dimensional lattice with density $p$.
I am able to follow the proof steps, and so I understand that the statement must be true. I also understand the example given, which is that if there exists an open path of edges between two points, it increases the probability of an open path between any other two points. This intuitively makes sense to me.
But when I think of integrating a probability measure with respect to the probability space, I am not understanding the geometric shape of the sets A and B that allows it to be possible that the measure of a subset of both sets is larger than the product of the measures of those sets. (Equality occurs with independence, so by definition, this case is fine). In the sense that I am not able to draw a picture of what this would mean. (Also I have shaky foundations in measure theory, so there may be an obvious answer here, but I don't have the background to see it, my apologies.)