In my book the Lebesgue integral is defined, with no reference to measure, as follows:
and the authors proceed to show that the integral is well-defined in the sense that if two different sequences of step functions, $f_n$ and $g_n$ satisfy (a) and (b) then $\sum_{n=1}^{\infty} \int f_n=\sum_{n=1}^{\infty} \int g_n$. This is fine but it is followed by a very confusing paragraph:
Could you please explain to me why this is the case based on what we know? In particular, I don't see why the function $f$ must vanish outside of $(a,b)$ and why the sequence cannot exist.
Thank you in advance.