Function: $f = -3\cdot 1_{[-2,-1]}+2\cdot1_{[3,+\infty)}$ (It is a step function)
If figured out that the integral wrt. the Lebesgue measure is equal to \infty as
$f= -3\cdot(2)+2\cdot(\infty)=\infty$
I need to figure out if the function $f$ belongs to any Lp space, for $p>1$. I know that it does not belong in any Lp space for $p<\infty$. But I am not sure if it belongs to the space for $p = \infty$
Your integral is wrong, it should rather be $$\int_{\mathbb R} f\,\mathrm d \lambda =-3+2\cdot \infty =\infty .$$
Moreover, $$|f|=3\boldsymbol 1_{[-2,-1]}+2\boldsymbol 1_{[3,\infty )}\geq \boldsymbol 1_{[3,\infty )}.$$
Therefore $$|f|^p\geq \boldsymbol 1_{[3,\infty )}\notin L^p(\mathbb R),$$ for all $p\geq 1$.