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Let $\mu (X)=+\infty$ and $p\in [1,\infty]$. I'm considering counterexamples about the inclusion of $L_p$ spaces, in particular with $L_\infty$. Taking $$f=1 \;\text{with}\;X=\mathbb{R}$$ it's clear that $f\notin L_p$. On the other hand, I'm looking for a function $$f\in L_p \;\text{but}\; f\notin L_\infty$$ For a positive valued function, $f:X\longrightarrow [0,+\infty]$, we know that if $f^p$ is integrable then $f^p$ must be finite a.e., therefore $f$ must be finite a.e. So, $f\in L_\infty$. Isn't it?

In a general case where $f:X\longrightarrow\mathbb{C}$ how can I find a counterexample? Or still true that $f\in L_p\Longrightarrow f\in L_\infty$?

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    $\begingroup$ For $X=\mathbb R$ with the Lebesgue sets and measure the usual counter-example is $$f_p(x) = \mathbf 1_{]0,1[}(x) x^{-1/(2p)}.$$ We have $f_p\in L^p$ but $f_p\not\in L^\infty$ for all $p\in[1,\infty[$. In fact this even works on $X=[0,1]$ with the Lebesgue measure. $\endgroup$
    – Maximilian Janisch
    Commented Nov 16, 2021 at 17:09

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