Ex3-9

Chapter 3 Question 9

Let $$\{E_k\}_{k=1}^{\infty}$$ be a sequence of sets so that $$\sum|E_k|_e$$ is finite. Then for any positive number $$\epsilon>0$$, there exists an integer $$j>0$$ such that

$$\begin{align}

\sum_{k=j}^{\infty}\left|E_k\right|_e<\epsilon

\end{align} $$

From Theorem 3.4, we have

$$\begin{align}

\left|\bigcup_{k=j}^{\infty} E_k\right|_e\leq\sum_{k=j}^{\infty}\left|E_k\right|_e<\epsilon

\end{align}$$

From the definition of $$\lim \sup$$ $$\begin{align}

\lim \sup E_k=\bigcap_{i=1}^{\infty}\left(\bigcup_{k=i}^{\infty}E_k\right)

\end{align}$$

gives us that for each $$i$$,

$$\begin{align}

\lim \sup E_k\subset \bigcup_{k=i}^{\infty} E_k

\end{align}$$

Then from the above part, we know for each $$\epsilon>0$$, there is a $$j$$ such that

$$\begin{align}

\left|\bigcup_{k=j}^{\infty} E_k\right|_e<\epsilon

\end{align}$$

which implies

$$\begin{align}

\left|\lim \sup E_k\right|<\epsilon.

\end{align}$$

Since $$\epsilon$$ can be arbitrary small, we have proven that $$\lim \sup E_k$$ has measure zero.

Since

$$\begin{align}

\lim \inf E_k \subset \lim\sup E_k

\end{align}$$

we can also get that $$\lim \inf E_k$$ has measure zero