$L^p$ spaces in integration measure

Question

This question looks simple at the first glance but ... I have tried to combine the theorems and definitions on $L^p$ spaces to solve this question but I have not been able to do so. I need help to show that, there is a measurable function $$g\in L^ { p_{0}}\setminus \Bigg( \bigcup_{p\neq {p_{0}}}L^p \Bigg)$$ for every $ 0<p_{0} < \infty$.

Answer 1

The answer for $p\geqslant 1$ is given in this theread. Given a $p>0$, take a function $f$ which is only in in $L^{Np}$ where $N$ is such that $Np\geqslant 1$. Then consider $g:=|f|^{1/N}\in L^p$ but not in $L^q$ if $q\neq p$.