Hi. I have a proof of a very basic measure theory theorem related to the definition of a measure, and would like to ask posters if the proof is wrong.(adsbygoogle = window.adsbygoogle || []).push({});

Theorem: If [itex]E[/itex] is measurable, then [itex]\overline{E}[/itex] is measurable and conversely.

My Proof:

Let's try the converse version first.

[itex]m(E)=m(E \cap \overline{E})+m(E \cap E)[/itex]

[itex]=m(E \cap \overline{E})+m(E)[/itex]

So [itex]m(E \cap \overline{E})=0[/itex]. By this we've shown that [itex]\overline{E}[/itex] is measurable. Converse is true by similar method.

[itex]m(\overline{E})=m(\overline{E} \cap \overline{E})+m(\overline{E} \cap E)[/itex]

[itex]=m(\overline{E})+m(E \cap \overline{E})[/itex]

[itex]=m(\overline{E})+0=m(\overline{E}).[/itex]

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# My proof of very basic measure theory theorem

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