Use the tower law and Jensen (conditional version).
Jensen:
Tower law:
I want the conditional expectation E[X|X>c] to be finite. I know X>=0. Is E[X]<infinity sufficient or do I need additional conditions? I'd think I need more (bounded second monent?), but I can't come up with an example for which E[X] is finite and E[X|X>c] is infinite.