I want to take the Lebesgue integral of the following function over $\mathbb R$, where $1\le p\lt q\le \infty$.
$f(x)=\begin{cases} x^{-1/p}, & x\ge1 \\[2ex] 0, & x<1 \end{cases}$
And I have that:
$\int_\mathbb R|f|^qdm=\int_1^\infty|f|^qdm=\int_1^\infty x^{-q/p}dx=[\frac1{1-q/p}x^{1-q/p}]_1^\infty=\frac{-1}{1-q/p}$
How, exactly, does the evaluation at "$\infty$" disappear?
\begin{align*} (L)\int_{1}^{\infty}x^{-q/p}dx&=\lim_{N\rightarrow\infty}(L)\int_{1}^{N}x^{-q/p}dx\\ &=\lim_{N\rightarrow\infty}(R)\int_{1}^{N}x^{-q/p}dx\\ &=\lim_{N\rightarrow\infty}\dfrac{N^{1-q/p}}{1-q/p}-\dfrac{1}{1-q/p}\\ &=-\dfrac{1}{1-q/p}, \end{align*} where $(L)$ stands for Lebesgue integral, $(R)$ stands for Riemann integral, and the first equality comes from Monotone Convergence Theorem.