4
$\begingroup$

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, $f \in L^p(X)$.

2 Answers 2

3

HINT

For part $(a)$, consider $B_1=\bigcup_{n=1}^{m}A_n$ such that $\mu(B_1) > 1$. This can be done since $\mu(\bigcup_{n=1}^{\infty}A_n) = \infty$ (Why?). Then define $B_{k+1} = \bigcup_{l=1}^{m_k}A_{l} \backslash B_k$ choosing $m_k$ such that $\mu(B_{k+1}) \geq 1$.

For part $(b)$, when $X = (0,\infty)$, we can consider $f(x) = \dfrac1x$. You can generalize this to any $\sigma$-finite space $X$. Once you obtain your $B_k$'s, let $f = \dfrac1{k\mu(B_k)}$ on the set $B_k$.

  • 0
    That's much better...2012-06-13
0

take $X=\mathbb{R}_+$ with normal topology the first cover could be $A_n=[n-1,n+1]$ hence $ A_n \cap A_{n+1}=[n,n+1] $ and then $B_i$ could be $ [n,n+1)$ and for $f$ take $ f(x)=\frac{1}{x}$ then $\int_{0 }^{\infty} |f(x)| = \infty$ and for any $p>1$ $\int_{0 }^{\infty} |f(x)|^p$ is finite.

  • 0
    Sorry, I thought the homework was about the abstract case so I tried to provide an example, but then again the example may lead in an straight forward way to a solution of the homework...2012-06-13