How much does the $L^p$ norms say about a function?

Question

Let's say we have two positive, decreasing function $u$ and $v$ on $[0,+\infty)$, and we know that $\|u\|_{L^p}=\|v\|_{L^p}$ for all $p\ge1$, can we say something about $u$ and $v$? Do they have to be the same function?

Answer 1

I did not check all details, but you can try this: For the Lebesgue measure $\lambda$ on $[0,\infty)$ an application of Fubini's theorem gives,for $p>1$, $\|u\|_{L^p}^p=p \int\limits_0^\infty x^{p-1} \lambda(\lbrace u>x\rbrace)dx$. Then approximate $f(x)= \lambda(\lbrace u>x\rbrace)- \lambda(\lbrace v>x\rbrace)$ by ploynomials to get $\int_0^\infty|f(x)|dx=0$ and hence $f=0$ by continuity. From $\lambda(\lbrace u>x\rbrace)= \lambda(\lbrace v>x\rbrace)$, continuity, and the monotonicity of $u,v$ it should follow that $u=v$.