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

Question

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.

Answer 1

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 $r$.