# $L_\infty$ inclusion

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 $$f\in L_p\Longrightarrow f\in L_\infty$$?

• 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. Commented Nov 16, 2021 at 17:09