Lebesgue Outer measure and Continuity from below

By | December 12, 2015

Lebesgue outer measure인 경우에는 continuity from below가 measurability에 상관없이 항상 성립한다.

Problem. If E_{1}\subset E_{2}\subset E_{3}\subset\cdots, then

    \[ m^{*}\left(\bigcup_{k=1}^{\infty}E_{k}\right)=\lim_{k\rightarrow\infty}m^{*}\left(E_{k}\right). \]

Proof. If m^{*}\left(\bigcup_{k=1}^{\infty}E_{k}\right)=\infty, then \sup_{k}\left\{ m^{*}\left(E_{k}\right)\right\} =\infty. So we may assume that m^{*}\left(\bigcup_{k=1}^{\infty}E_{k}\right)<\infty. Let A_{k} be a measurable hull of E_{k} and define B_{k}=\bigcap_{j=k}^{\infty}A_{j}.
Then note that B_{k} is also a measurable hull of E_{k} since E_{k}\subset E_{l}\subset A_{l} for any l\ge k. So E_{k}\subset\bigcap_{j=k}^{\infty}A_{j}.

    \[ m_{*}\left(\bigcap_{j=k}^{\infty}A_{j}\setminus E_{k}\right)\le m_{*}\left(A_{k}\setminus E_{k}\right)=0. \]

Also, B_{1}\subset B_{2}\subset B_{3}

So by continuity of measure, we have

    \[ m\left(\bigcup_{k=1}^{\infty}B_{k}\right)=\lim_{k\rightarrow\infty}m\left(B_{k}\right). \]

Then by fact, we have

    \[ m^{*}\left(\bigcup_{k=1}^{\infty}E_{k}\right)=\lim_{k\rightarrow\infty}m^{*}\left(E_{k}\right) \]

since \bigcup_{k=1}^{\infty}B_{k} is a measurable hull of \bigcup_{k=1}^{\infty}E_{k} and B_{k} is a measurable hull of E_{k}.

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