It's kind of a generalized pigeonhole principle, if we view the pigeonhole principle as an exercise in elementary measure theory, a.k.a. counting. Two disjoint measurable sets such as the ones below will always have a combined measure equal to the sum of their measures. If that's greater than the total measure of the space you're in, then the disjoint sets weren't. —Dan -----

...

My intuition says that since those two sets are each in the unit interval, which of course has measure 1, and since they each have a measure of more than 1/2, that by the pigeonhole principle overlaps are inevitable.