Lebesgue measure from $L^1(I)$ using metric completion? Given a closed cube $I \subseteq \mathbb{R}^N$ (e.g. $I = [-1,1]^N$), we can define a norm $\|\cdot\|_1$ on the vector space $\mathcal{C}(I, \mathbb{R})$ of continuous functions by $\|f\|_1 = \int_I |f(x)|dx$, where the integral is Riemann. This defines an incomplete metric vector space. One can show the completion is the Banach space $L^1(I)$ with the Lebesgue measure/integral.
I know that one can show this relationship directly by using properties of the Lebesgue integral, but my question is whether we can work in reverse: can we recover/define the Lebesuge measure on $I$ simply using the traditional construction of the Banach completion of $(\mathcal{C}(I, \mathbb{R}), \|\cdot\|_1)$ (i.e. equivalence classes of Cauchy sequences of continuous functions)?
So far my plan for answering this has been to first define a $\sigma$-algebra on $I$ based on the characteristic functions. That is, we'd informally say $E \subseteq I$ is measurable if $\chi_E : I \to \mathbb{R}$ is "an $L^1$ function", meaning that there is a Cauchy sequence $(f_k)\subseteq \mathcal{C}(I, \mathbb{R})$ that represents $\chi_E$ inasmuch as the norms converge to what should be $\int_E \chi_E d\mu$. That number would then be defined as the measure of $E$.
I think it should be relatively simple from there to show we indeed get a $\sigma$-algebra and a measure on it, but now the problem is to rigorously define what "is an $L^1$ function" means for $\chi_E$. For some subsets, this is easy, like $E = \varnothing$ and $E = I$. In those cases, $\chi_E$ is itself a continuous (constant) function. How should we proceed from here, if we can at all?
 A: In addition to the references that Oliver has suggested, I recommend taking a look at Chapter I of Vol I of Reed & Simon Functional Analysis.  Following Thm I.7 (Completion of Bounded Linear Transformations), Riemann Integral was defined, from the functional point of view, as a completion.  This is done only on the bounded piecewise continuous functions, $PC[a,b]$.  We are naturally led to the question of completeness of this norm on the space of continuous functions $C[a,b]$.  This discussion of Lebesgue Integral as completion is found on p12, though I have to quote the authors saying

We can always complete $C[a,b]$ in $\|\cdot\|_1$ realizing elements of the completion as equivalence classes of Cauchy sequences of continuous functions; this realization is not noteworthy for its transparency...

By the way, as we are on the topic of the space of continuous functions on $[0,1]$, we can ask the same question about the space of measurable functions on $[0,1]$ and the space of bounded functions on $[0,1]$.  This is partially mentioned in the introduction to measure theory and Lebesgue integral in Theory of Linear Operators written by Banach.  Ha!
