The question and the solution are given in the following quotes:
- Determine which of the following functions are in $L_2[0,1]$, then compute their norms.
(a) $\frac{1}{\sqrt{t^2+1}}$
(b) $\frac{\sqrt{a}}{\cos{at}}$, $a \in \mathbb R$
(c) $\frac{1}{t^a}$, $a \in \mathbb R$
(d) $t^n e^{at}$, $a \in \mathbb R$, $n \in \mathbb N$
Solution: (c). Proof:
$\int_{[0,1]}\lvert{\frac{1}{t^a}}\rvert^2 = \int_{[0,1]}\frac{1}{t^{2a}}$. If $2a \lt 1$, the improper Riemann integral exists and is absolutely convergent. Thus, $f(t)$ is also Lebesgue integrable and $\int_{[0,1]}\frac{1}{t^{2a}} = \int_0^1 \frac{dt}{t^{2a}} = \frac{t^{1-2a}}{1-2a}\rvert_0^1 = \frac1{1-2a} \lt \infty$, which implies that $\int_{[0,1]}\lvert{\frac{1}{t^a}}\rvert^2 = \infty$. Thus, $\frac1{t^a} \in L_2[0,1]$ iff $a \lt \frac12$.
But I did not understand why the improper Riemann integral exists and is absolutely convergent. When $2a \lt 1$, and if $2a \geq 1$, why is it divergent?
Also, I did not understand how the author concluded that $f(t)$ is also Lebesgue integrable. Could anyone tell me which rule the author used to show this?
