Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Question

Let $(X,A,μ)$ be a $σ$-finite measure space with $μ(X) = ∞.$ Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

I could easily do this if I get to choose what the space is. How do I construct a function for an arbitrary space? Thanks.

Answer 1

Take an exausting sequence of measurable sets $X_n\subset X_{n+1}$, $n=1,2,\dots$ such that $\bigcup_n X_n=X$, $\mu(X_n)<\infty$ and $\mu(X_{n+1})-\mu(X_n)\ge1$. Set $Y_n=X_{n+1}\setminus X_n$ and define $F(x)=(n\mu(Y_n))^{-1}$ for $x\in Y_n$.

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The requirement that $\bigcup X_n=X$ is of course superfluous; it suffices that $\mu(\bigcup X_n)=\infty$, and we can set $F(x)=0$ on the rest of $X$.