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I am trying to get a better understanding of the definitions of Lebesgue inner and outer measures. Let $E$ be an arbitrary set of real numbers, and for any open interval $I=(a,b)$ define $\lambda(I)=b-a$.

  • The Lebesgue outer measure $\lambda^\ast$ is defined as $$ \lambda^\ast(E) = \inf\{\lambda(G):E \subset G, G \ \text{is open}\}. $$
  • The Lebesgue inner measure is defined as $$ \lambda_\ast(E) = \sup\{\lambda(F):F \subset E, F \ \text{is compact}\}. $$
  • If these measures of $E$ are equal we say that $E$ is Lebesgue measurable with the Lebesgue measure $\lambda(E) = \lambda^\ast(E) = \lambda_\ast(E)$.

Now lets say I have the half open interval $E=[1,3)$. Clearly the measure of this set is $2$. But How can I formally use the definitions above to calculate $\lambda^\ast(E)$ and $\lambda_\ast(E)$ and show that they equal $2$?

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I think that there is something missing in your statement, since you define $\lambda$ only on open intervals, but you do not say how it is defined on open sets or compact sets. One usually defines the Lebegue outer measure as

$$\lambda^*(E) = \inf \sum_n |Q_n|,$$

where the infimum is taken over countable covers of $E$ with closed cubes, and $|Q_n|$ is the elementary geometric volume. A set is said to be Lebegue measurable if for all $\varepsilon >0$ there is an open set containing $E$ with $\lambda^*(O - E) \le \varepsilon$ and for such sets the Lebegue measure is defined as the exterior measure.

Edit:

Okay, then $E \subset (1-1/n,3)$ and $\lambda(1-1/n,3)=2+1/n$, hence $\lambda^*(E) \le 2$. Moreover, $[1+3-1/n] \subset E$ and $\lambda([1+3-1/n])=2-1/n$ (this is already a closed interval), hence $\lambda_*(E) \ge 2$ and you are done since $\lambda_* \le \lambda^*$ (the smallest closed interval containing a compact set is contained in the smallest open interval containg that set).

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  • $\begingroup$ I didn't put in all the background information as the post would be too long, but the definitions you are asking about are given on page 102 of this book - classicalrealanalysis.info/documents/… - and page 104-105 contain the definitions I used in my post. $\endgroup$ – eurocoder Jul 6 '17 at 6:59

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