In my analysis class, I was given the following definition of the Lebesgue integral:
Let
$O^b:=\{ \phi:[a,b]\to\mathbb R \mid \exists \text{ sequence of step functions $s_1 \leq s_2 \leq \ ... \ \rightarrow \phi$ a.e.}\}$
$O^a:=\{ \phi:[a,b]\to\mathbb R \mid \exists \text{ sequence of step functions $s_1 \geq s_2 \geq \ ... \ \rightarrow \phi$ a.e.}\}$
On $O^b$ define $\int^b_a \phi:= \lim \int^b_a s_n \leq +\infty$
On $O^a$ define $\int^b_a \phi:= \lim \int^b_a s_n \geq -\infty$
$L$-$\overline{\int^b_a}f:= \inf \{ \int^b_a \phi \mid f\leq \phi \in O^b \} $
$L$-$\underline{\int^b_a}f:= \sup \{ \int^b_a \phi \mid f\geq \phi \in O^a \} $
$O^b$ and $O^a$ stand for One-sided$^{\text{below}}$ and One-sided$^{\text{above}}.$
If $L$-$\underline{\int^b_a}f = L$-$\overline{\int^b_a}f$, we call it the Lebesgue integral of $f$.
In order to get a better understanding of what's going on, I tried to look up the definition in a textbook or on the internet but I couldn't find anything (not even close to the above). The problem is that we didn't have any measure theory so far: only the zero set (hence the "a.e." above). And all texts seem to define the Lebesgue integral after introducing measure etc.
I would highly appreciate a reference to a material which introduces the Lebesgue integral in the above way.
Look in Pesin, I.N. Classical and modern integration theories. (English) Probability and Mathematical Satistics. 8. New York-London: Academic Press. XVIII, 195 p. (1970). MSC 2010: 26-01 28-01 26A36 26A39 26A42 28Cxx 26-03 28-03 Sorry, I can't add a comment.