Is it possible to have a function that is in $L^{\infty}(T)$ but not in $L^1(T)$?

Question

I know that there are functions that exist in the $L^1$ norm but not in the $L^{\infty}$. I'm having trouble coming up with the converse. Could someone provide me an example?

Answer 1

If $T$ is of finite measure $|T|<\infty$, then $$ \int_T |f| ≤ \|f\|_{L^\infty(T)} \int_T 1 = \|f\|_{L^\infty(T)}\,|T|. $$ Otherwise, if $|T|=\infty$, the function $f=1$ is a good counterexample since $$ \|1\|_{L^\infty(T)} = 1 \ \text{ but }\ \|1\|_{L^1(T)} = \int_T 1 = |T| = \infty $$