Let $f(x) = x^{-\alpha}, ~ x \in (0,1], ~ \alpha \in \mathbb{R}$, then how can we show that $f$ is Lebesgue integrable?
I can show that $f$ is measurable but I don't know how to proceed further. I think I need to use monotone convergence theorem to find a sequence such that integral will be finite, but how can I construct such a sequence?
Your function is integrable iff $1-\alpha >0$ or $\alpha <1$.
Let $f_n(x)=x^{-\alpha}$ for $x >\frac 1n$ and $0$ for $x \leq \frac 1 n$. The integral of $f_n$ is actually a Riemann integral and it can be evaluated explicitly. Now apply Monotone Convergence Theorem to see when $f$ is integrable.
[$(f_n)$ is sequence of non-negative measurable functions increasing to $f$ at every point. Hence $\int_0^{1}f=\lim \int_0^{1} f_n=\lim \int_{1/n}^{1} f_n=\lim x^{1-\alpha}|_{1/n}^{1}$. I will let you finish the argument].