I have this exercise:
Let $(\Omega,\mathcal{A},\mu)$ be an arbitrary measure space and $1 <p <\infty$. Show that if $l \in \mathcal{L}^p(\mu)^*$, then there exists a sequence $\{\Omega_n\}_{n=1}^\infty$ of $\mathcal{A}$-measurable sets such that $\mu(\Omega_n)<\infty$ for each $n \in \mathbb{N}$ and $l(\mathcal{X}_A)=0$ for each $A \in \mathcal{A}$ such that $\mu(A)<\infty$ and $A\subset(\cup_{n=1}^\infty\Omega_n)^c$.
I have no idea of how to define the set $\Omega_n$. My first idea was $\Omega_n=\bigcup\{A | \mu(A)<\infty, |l(\mathcal{X}_A)|<\frac{1}{n}\}$. But two problems arise here, I do not know if the set defined has finite measure, and if K is a subset of that set, it doesnt follow(easily?) that $|l(\mathcal{X}_K)|<1/n$, or is bounded by a number. But the least requirement follows, if a set is in none of the $\Omega_n$, then the linear functional must send the relevant characteristic function to $0$.
So it seems that what I tried made the sets $\Omega_n$ too big.
Can you guys please help?