This answer (https://math.stackexchange.com/a/1540107/273275) uses $$f(x) = x^{-p},\qquad x\geq1$$ is lebesgue integrable for $p>1$. But I did not manage to prove it. $f$ is obviously measurable. To show integrability, one has to find a sequence of simple functions which conerge to $f$.
Since I am not allowed to use any Riemann integral stuff, I have to prove integrability by constructing a clever simple function (or use some theorems like Monotone Convergence).
So far I thought about using $s_n(x) = f(x)\chi_{[1,n]}(x)$. Since this function is positive, I can use Monotone Convergence to show that $\int f = \int\lim s_n = \lim\int s_n = \lim\int_1^nx^{-p}$ but how do I know — without using Riemann stuff — that the latter integral is finite? I tried to find an upper bound by $\int_1^nx^{-p} \leq \sup_{x\in[1,n]} x^{-p}\cdot\lambda([1,n]) = n - 1$ which is = $\infty$ for $n$ approaching infinity. Is there are smarter choice of $s_n$?
Just notice $f(x) = x^{-p} \leq n^{-p}$ if $x\in (n,n+1]$. So we have $$\int_1^n f(x) dx = \sum_{k=1}^{n-1} \int_{k}^{k+1}f(x)dx \leq \sum_{k=1}^{n-1} k^{-p} \leq \sum_k k^{-p}<\infty.$$
Therefore, $$\int_1^\infty f(x)dx = \lim_n \int_{1}^n f(x)dx \leq \sum_k k^{-p}<\infty.$$