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

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 $x\in Y_n$.

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 $X$.
