(Lebesgue integral) Can simple functions take on infinity?

But your umption was that phi <= f, if phi becomes unbounded then surely Int f du becomes infinite and the supremum{Int phi du | phi} = infinity, so in that sense it's still true. That's what I'd think, but it might be better to consult a text such as Mathematical ysis by Rudin. Unfortunately I'm strictly an amateur at this sort of thing...