I assume chi is the indicator function, allow me to use the bolded one. Assume that where is an interval partition of . Assume that is not a pairwise-disjoint set. Then, we may assume (with the possibility of relabeling) that . Show that this implies that .

If this was unhelpful, in the sense that I didn't really say anything substantiative, then just tell me and I'll try to give a better hint.