Measure theory $L^p$ and $L^q$ spaces

Question

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$.

Idea:

This and the function $f=x^{-1/p}(1+|log x|)^{-2/p}$ and then I need to do the integration $\int|f|^p dm$ and $\int|f|^qdm$ but I don't understand why. Any help?

Answer 1

You can compute the improper integrals to show that $\displaystyle \int_0^1 x^{-\alpha} \, dx < \infty$ if and only if $\alpha < 1$, and $\displaystyle \int_1^\infty x^{-\alpha} \, dx < \infty$ if and only if $\alpha > 1$.

Assume without loss of generality that $1 < p < q < \infty$. Then $x^{-1/q} \chi_{(0,1)}$ belongs to $L^p \setminus L^q$, and $x^{-1/p} \chi_{(1,\infty)}$ belongs to $L^q \setminus L^p$.