Is $L^\infty$ a subset of $L^p$ for finite $p$?

Question

If we have a probability measure, then can we say that $L^{\infty}(\mu) \subset L^{P}(\mu)$?

If a function is in $L^{\infty}(\mu)$ then it is essentially bounded but bounded doesn't necessarily imply that $\int_{X} |f|^p d\mu < \infty$, correct? But if this is the case then I am not sure where else to begin...

Answer 1

For a probability measure or any other finite measure this is correct.

If $|f|$ is bounded by $C>0$ a.e., you have $$ \int_X |f|^p \mathrm d\mu \leq C^p\int_X \mathrm d\mu \leq C^p \mu(\Omega) < \infty $$