A practical method to determine if $f \in L^p(\mathbb{R})$

Question

I know these definitions:

L-integrable function:

$f: \mathbb{R}^n \rightarrow \mathbb{C}$ is L-integrable if there is a scalable functions succession $s_n$ that:

$$\lim_{n\rightarrow +\infty} s_n(x)=f(x) $$ $$\forall \epsilon>0 \qquad \exists \ n^* \qquad\forall m,n\ge n^* \qquad \int_{\mathbb{R}^n} \rvert s_n(x)-s_m(x) \rvert < \epsilon$$

and $L^p(\mathbb{R})=\{ x \in \mathbb{R} : \rvert f \rvert^p$ is L-integrable $ \} $

Is there any practical method to determine if $f\in L^1(\mathbb{R})$ or $f\in L^2(\mathbb{R})$?

Thanks!

Answer 1

$$\forall p\in \mathbb{N}^*,\quad f\in L^p(\mathbb{R})\Longleftrightarrow\parallel f\parallel_{L^p(\mathbb{R})}<\infty$$ Where, $$\parallel f\parallel_{L^p(\mathbb{R})}=\left(\int_\mathbb{R}|f(x)|^pdx\right)^\frac{1}{p} $$