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How do I show that the set $[0,1]$ is measurable? I intend to use the usual definition of measurable sets i.e A is measurable if $\mu^{*}(E)=\mu^{*}(E\cap A)+\mu^{*}(E\cap A^{\complement})$

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You need to enclose the $\LaTeX$ in single dollar signs to get it to display properly. Using double dollar signs will give you displayed formulas. –  Brian M. Scott Jan 27 '13 at 0:07
are you following any textbook on measure theory? any such text will include a proof of this result. –  Ittay Weiss Jan 27 '13 at 0:08
I am following the book [Gerald B.Folland: Real Analysis Modern Techniques]. There is no proof for this particular question. Another question is: Is $(0,1)$ measurable as well? I was thinking of considering $E\subset \mathbb{R}$ If $E\cap [0,1]=\emptyset$ then $\mu^{\ast}(E)\leq\mu^{\ast}(E\cap [0,1]^{\complement})\leq\mu^{\ast}(E)$ hence measurable. am not sure how to proceed if $E\cap [0,1]\neq\emptyset$. –  obiero Jan 27 '13 at 21:08
... and you think Folland's (1.16) isn't enough? –  GEdgar Jan 27 '13 at 22:12

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