I'm trying to find some example for those two question. But it's not familiar for me, so hard to find. Please some Hint for those.
Suppose $\mu(X) = \infty$ and there exist sets $A_1, A_2, A_3, \dots$ in $M$ such that $\mu(A_k) < \infty$ for every $k$ and $X= \bigcup\limits^\infty_{k=1} A_k$. ($X$ is $\sigma$ -finite)
(a) Prove that there exist disjoint sets $B_1, B_2, B_3, ...$ in $M$ such that $1\le \mu(B_k) < \infty$ for every $k$ and $X= \bigcup\limits^\infty_{k=1} B_k$.
(b) Prove that there exists $f \notin L^1(X)$ such that for all $1
, $f \in L^p(X)$.