I have n boys and n girls. I need to pair as much of them as possible for a dance in O(nlogn). Reduce this to a standard problem?
Algorithm: Pick girl #i where $P_i$ is as high as possible. Eligible boys are those with $A_j ≥ P_i ≥ B_j$. If there is no eligible boy then she is not matched, otherwise match her with eligible boy #j, where $B_j$ is as large as possible. Then match the remaining boys and girls in an optimal way. 

If instead you match her with boy #k or with nobody, then you can just swap boy #j and boy #k, and you get a result either as good as promised.