In the paper "A uniform algebra of analytic functions" of Carne-Cole-and-Gamelin. The Theorem 6.4 says:
Theorem 6.4: Let $\mu$ be a positive measure on some measure space, let $2\leq p<\infty$, and let $N$ be a integer satisfying $N\geq p$. Let $\{f_j\}$ be a sequence in $L_p(\mu)$ such that $f_j\longrightarrow 0$ weakly, and $F(f_j)\longrightarrow 0$ for every $F\in\mathcal{P}_N(L_p(\mu))$. Then $\|f_j\|_p\longrightarrow 0$.
Here, $\mathcal{P}_N(L_p(\mu))$ is the space of N-homogeneous polynomials.
Proof: Since the $f_j$'s are all carried by some $\sigma$-finite set, we can assume that the measure $\mu$ is $\sigma$-finite. Replacing $\mu$ by a mutually absolutely continuous measure, we can assume that $\mu$ is a probability measure. $\cdots$
My questions. Someone can help me understand this change of a arbitrary measure space to a probability measure space? Why the functions are carried by a $\sigma$-finite set?
Thanks for any hint.
Let $A_{j,n} = \{ |f_j| > 1/n\}$. Since $f_j \in L^p$, each $A_{j,n}$ must have finite measure. So if we let $A = \bigcup_{j,n} A_{j,n}$, then $A$ is $\sigma$-finite. And every $f_j$ vanishes outside $A$. This is what they mean by "all $f_j$'s are carried by some $\sigma$-finite set."
I'll leave the part about changing to a probability measure for someone else.