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Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space.

Is $L^\infty(\mu)\subseteq L^p(\mu)$ for all $p\ge 1$?

I know there is another question about the inclusions of the $L^p$-spaces, but the answers don't cover the case above.

It's clear that if $\mu(\Omega)<\infty$, then $L^\infty(\mu)\subseteq L^p(\mu)$ for all $p\ge 1$. Clearly, if $f\in L^\infty(\mu)$, then $$|f|\le C\;\;\;\mu\text{-almost everywhere}$$ for some $C\ge 0$ and hence $$\left\|f\right\|^p_{L^p(\mu)}\le C^p\mu(\Omega)<\infty\;.$$ I guess that $L^\infty(\mu)\not\subseteq L^p(\mu)$ in general, but wasn't able to find a proper counterexample.

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The Lebesgue measure on $\Bbb R$ is a counterexample. The non-zero constants are not in any $L^p$, but they are of course in $L^\infty$. In fact, this happens as soon as $\mu(\Omega)=\infty$.