It is well known that the completion of $C[0,1]$ with the norm given by $\|f\|=\int_0^1|f|$ is $L^1(0,1)$ and that the extension of the Riemann integral functional to $L^1(0,1)$ is the Lebesgue integral. I want to teach the Lebesgue integral in $(0,1)$ without using measure theory neither the Daniell Integral. I want some practical (calculus) tools to compute Lebesgue integrals and to decide when a function is in $L^1(0,1)$ or not. I want to use only the concept of zero measure sets.
Denote with $$\mathcal L^1(0,1)=\{f \,:\,\exists f_n \in C[0,1] \text{ Cauchy and } f_n(x)\to f(x) \text{ a.e. }(0,1) \}/\text\{f\,:\, f(x)=0 \text{ a.e. }(0,1)\}.$$ I have done some computations and I think it is possible to prove that $\mathcal L^1(0,1)$ is the same thing that $L^1(0,1)$ with its natural identification, but I am not happy with those computations. Some of them are a little bit messy.
¿Is there some text that develop those ideas? I don't care if it is heavy reading or it has an abstract configuration.