N90 Controller providing OJT described an event involving a loss of separation event; noting the event occurred because of attempts to let the trainee go as far as possible for needed experience.

Narrative

We were training on final. There was a lot going on; traffic was very busy; there was coordination going on regarding an emergency; the Trainee was vectoring to try to stay in our airspace; trying to make a point out; etc. Basically; the Trainee created too many different areas to try to focus on and was unable to see the MD88 was starting to overtake the B752. I thought the trainee had already switched the MD88 to Tower but during the replay I learned that was not the case. The Tower broke the MD88 out and from looking at it on our RADAR scope; it appeared that they canceled the approach prior to losing 4 NM; but the Quality Assurance personnel said spacing got down to 3.9 NM or something like that. I'm responsible for the position; so ultimately it's my fault. I let the Trainee go too far and I didn't fix the overtake situation. It's a tough situation because the Trainee is having difficulties and I try to let him/her go as far as possible and we're really working on trying to make him/her tell the feeders when to slow and go into a hold; and in this case; because I thought the Tower was working the two aircraft involved; and we had a lot of other stuff going on at the same time; I didn't see/fix the overtake. Even though I think it's kind of a bogus deal; (3.9 instead of 4); I don't take it lightly and it's the first operator error I've had in many years. I'm more mad that I didn't catch it and fix it than anything.