$L^\infty$, $L^p$ question

Question

Suppose $f \in L^0$. I read that for a general measure space, if $\mu(X)<\infty$, then we cannot have that both $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$, but if $\mu(X)=\infty$, then we can find an $f$ such that this statement is true. I'm not sure how to show this for the finite case and am having trouble thinking of an example of a function $f$ for the infinite case such that $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$.

Answer 1

Hint:

For the first case, you can actually show that

$$||f||_p \leq ||f||_\infty \mu(X)^{1/p}$$

if $\mu(X) < \infty$. For the second one, consider $X = \mathbb R$ with the Lebesgue measure. Can you found a bounded positive function $f$ on $\mathbb R$ so that

$$\int_\mathbb R f dx = + \infty?$$

(Don't think too hard)