I came across the example "Show that $\int _{0}^{1}x^{-a}dx$ exists as a Lebesgue integral, and is equal to $1/(1-a)$, if $0 < a < 1$; but is infinite if $a\geq 1$. The Lebesgue definition of the integral is $\lim _{n\rightarrow \infty }\left\{ \int _{0}^{n^{-\frac {1} {a}}}ndx+\int _{n^{-\frac {1} {a}}}^{1}x^{-a}dx\right\} $ and the results are the same as in the elementary theory.
To put my question bluntly i just do not understand why and how the author determined to split the original integral. I am aware if we take the limit the first integral's upper bound would become 0 and the second integral's lower limit would be 0 and the second integral would look the same as the one we were originally presented with. I suppose i do not quite understand the motivation behind the step. Any light shed on this matter would be much appreciated.
Edit: As per request the definition provided in the book is.
The Lebesgue integral of $f(x)$ over $(a, b)$ is the common limit of the sums $s$ and $S$ when the number of division-points $y_v$ is increased indefinitely, so that the greatest value of $y_{v+1} - y_v$ tends to zero. where $s=\sum _{v=0}^{n}y_{v}\mu\left( e_{\nu }\right) $ and
$S=\sum _{v=0}^{n}y_{v+1}\mu\left( e_{\nu }\right) $