I have two real functions $f$ and $g$ defined by $ f(x) =\begin{cases} \tfrac1x & \text{if } x \in (0,1] \\ 0 & \text{if } x \in \mathbb R\setminus (0,1] \end{cases} \qquad \text{and} \qquad g(x) = \begin{cases} \tfrac1x & \text{if } x \in (1,\infty] \\ 0 & \text{if } x \in (-\infty,1] \end{cases}.$
I have to calculate $\{p \in [0,\infty] \colon f \in \mathcal L^p(\lambda)\}$ and $\{p \in [0,\infty] \colon g \in \mathcal L^P(\lambda)\}$.
So far, I've thought this: Since $\mathcal L^\infty(\lambda)\subseteq\mathcal L^r(\lambda)\subseteq\mathcal L^0(\lambda)$, where $r \in(0,\infty)$, I only have to calculate $\mathcal L^0(\lambda)$, which is defined like this: $\mathcal L^0(\lambda)={f\in\mathcal M(\mathcal E): \lim_{t\to \infty}\lambda({|f|\ge t})=0}$. I am, however, not at all sure how to calculate this. I would appreciate help a lot.