Let $I = [0,1]$. Suppose we define $A$ as the space of all functions $f:I\to \mathbb{R}$ which are Riemann integrable. We can put a norm on this space $A$, by defining $||f|| = \int_I |f|$. Of course, this is not quite a norm, and we identify $f\sim g$ iff $\int_I |f-g| = 0$. The space $B = (A/\sim)$ will then we a normed function space.
Unfortunately, $B$ is not complete. However, we can complete it, and let us call the new resulting space $C$ (this is a Banach space). In classical real analysis we typically call this $L^1([0,1])$. However, we completely bypassed the construction of the Lebesgue integral.
We can then say $E\subseteq [0,1]$ is a "measurable" set if and only if $\chi_E\in C$ (the indictator function). We also define the Lebesgue measure of $E$ to be the quantity $||\chi_E||$, where $||\cdot ||$ is the extended norm to the Banach space $C$.
I never seen books take this approach to real analysis. This non-standard approach seems to be a lot simpler. Perhaps, there are some issues with this?
