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Let $(X, Ω, μ)$ be a finite measure space.

Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

I am having trouble starting this problem. It is the second part to another question on a past qual. Any help would be awesome. Thanks.

kingkongdonutguy
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    I would try something like this: for each sequence $t_n\to 0$, you can find a sequence $E_n$ such that $0<\mu(E_n)<t_n$ and $E_n\cap E_m=\emptyset$ for $n\neq m$. Let $a_n\to 0$ and consider the function $$f(x)=\sum_{i=1}^\infty a_n\chi_{E_n}(x)$$ I think it is possible to find $t_n$ and $a_n$ satisfying your thesis. – Tomás Aug 22 '14 at 00:37
  • How do we know we can choose the sets so the intersetion is empty? – kingkongdonutguy Aug 22 '14 at 00:46

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We can use Tomás' idea: define $$f(x)=\sum_{j=1}^{+\infty}b_n\mu(A_n)^{1/p}\chi(A_n)$$ where the sets $A_n$ are such that $\mu(A_n)\in (0,t_n)$ and $b_n$ is positive. If $\sum_{n=1}^{+\infty}b_n$ is convergent, then $f$ belongs to $\mathbb L^p$. We have to choose $t_n$ such that the sequence $(b_n/t_n^r)_{n\geqslant 1}$ does not converge to $0$ for each positive $r$.

Davide Giraudo
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