Jabarkan ||f|| = ∫ |f(x)| dari Integrasi Lebesgue dalam bahasa sunda

Then Cc is a normed vector space (and in particular, it is a metric space.) All metric spaces have completions, so let L1 be its completion. This space is isomorphic to the space of Lebesgue integrable functions (modulo sets of méasure zero). Furthermore, the Riemann integral ∫ defines a continuous functional on Cc which is dense in L1 hence ∫ has a unique extension to all of L1. This integral is precisely the Lebesgue integral.

In this formulation, the limit taking théorems are hard to prove. However, in more general cases (such as when the functions, or perhaps the méasures, take values in a large vector space instéad of Rn) this approach is a fast way of obtaining an integral.