A uniformly continuous linear extension principle in topological vector   spaces with an application to Lebesgue integration
  We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali Convergence Theorem and the Riesz-Fischer Theorem are shown to follow as easy consequences from our construction. 
Ben Berckmoes
math.FA