A simple proof of the Fundamental Theorem of Calculus for the Lebesgue   integral
  This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory. 
Rodrigo L\'opez Pouso
math.CA