This question is motivated by this one, which had no successful answer: Definition of measurable set as a limit of step functions
The most usual way to construct the Lebesgue integral is by means of measure theory. One starts with the concept of measurable function: given two $\sigma$-algebras $\mathcal{A}$ and $\mathcal{G}$, a function $f:(\Omega,\mathcal{A})\rightarrow (E,\mathcal{G})$ is said to be measurable if $f^{-1}(\mathcal{G})\subseteq \mathcal{A}$. In the real case, one takes $\Omega=\mathbb{R}^n$, $E=\mathbb{R}$ and $\mathcal{G}=\mathcal{B}(\mathbb{R})$ (the Borel sets). To define $\mathcal{A}$, one takes the Lebesgue outer measure, $$ \mu^*(A)=\inf\left\{\sum_{i=0}^{\infty} |I_i|:\, A\subseteq \bigcup_{i=1}^{\infty}I_i,\,\,I_i\text{ open rectangle}\right\},\quad A\subseteq\mathbb{R}^n, $$ and says that $A\in\mathcal{A}$ if and only if $A$ partitions every subset of $\mathbb{R}^n$ in a good way, that is to say, $\mu^*(B)=\mu^*(A\cap B)+\mu^*(A^c\cap B)$ for every $B\subseteq\mathbb{R}^n$. We say that $\mathcal{A}$ is the set of measurable sets in $\mathbb{R}^n$, and the outer measure $\mu^*$ restricted to the $\sigma$-field $\mathcal{A}$ is indeed a measure just denoted by $\mu$. From here, one can define the integral of $f:(\mathbb{R}^n,\mathcal{A})\rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$, $\int_{\mathbb{R}^n}f\,d\mu$, starting with simple functions, then with nonnegative functions and then in general.
However, there is another construction of the Lebesgue integral which needs nearly no measure theory (just the concept of null set). In $\mathbb{R}^n$, you consider the Lebesgue outer measure $\mu^*$ again. Then you can start defining the Lebesgue integral (yes, without even knowing what a measurable function is). One considers step funcions (funcions of the form $f=\sum_{i=1}^n a_i1_{I_{i}}$, where $I_i$ is a rectangle in $\mathbb{R}^n$), and the integral is defined as $\int f=\sum_{i=1}^n a_i |I_i|$. Then one moves to superior functions: $f$ is superior if it is an a.e. increasing limit of step funcions $\{f_i\}_{i=1}^{\infty}$ such that $\sup_n\int f_n<\infty$, and in such a case $\int f:=\lim_n\int f_n$. Finally, one says that $f$ is Lebesgue integrable if $f=g-h$, where $g$ and $h$ are superior functions, and $\int f:=\int g-\int h$. Then one can prove dominated convergence, Fubini... And now one defines a measurable function: a function $f$ such that it is an a.e. limit of step functions. A set $A$ is measurable if $1_A$ is measurable. In this way one can define the integral for general measurable functions: let $f$ be a measurable function; if $f^+,f^-\in L^1$, then $\int f=\int f^+-\int f^-$; if $f^+\not\in L^1$ and $f^-\in L^1$, the $\int f:=+\infty$, etc.
This second approach is followed, for example, in the book Mathematical Analysis, by T. M. Apostol.
I would like to know if both definitions of measurable set are equivalent, and if both "theories" are in fact equivalent, although in some sense one would be like the inverse of the other.