# Constructing the Lebesgue measure on the reals.

So far, we have only constructed the Lebesgue measure on intervals. Now if $\{I_k\}$ is a countable cover of the reals by intervals which overlap at one point at most, we say that $S\subset \mathbb R$ is measurable iff $S \cap I_k$ is measurable in $I_k$ for all $k$ and set $$m(S)=\sum_{k=0}^{\infty}m_{I_k}(S\cap I_k)$$ I have shown that the values coincide for any two covers, knowing that $S$ is measurable for all elements in both covers. How do we show that if S is measurable according to one cover, then it is measurable according to another? (Then showing that the family of measurable sets is a $\sigma$-algebra and additivity of the measure should pose little problem.)

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For two covers $\{I_k\}$ and $\{J_l\}$, use their intersections $\{I_k\cap J_l\}$, so that $I_k=\bigcup_l (I_k\cap J_l)$ and $J_l=\bigcup_k (I_k\cap J_l)$. –  Berci Feb 8 '13 at 16:09
What do you mean by "$S\cap I_k$ measurable in $I_k$"? You say you've defined measure only for intervals but the intersection isn't always an interval –  leo Feb 9 '13 at 19:27
We have constructed the Lebesgue measure on an interval. Not only subintevals are measurable. –  Student Feb 10 '13 at 10:06
@Berci Your suggestion is sound to show that the measure of a subset does not depend on the cover, given it is measurable according to both covers. If you read the question carefully, you will notice that this is not the issue. –  Student Feb 10 '13 at 10:16

You should show that if $I\subset J$ and $S\cap I$ is measurable in $I$ then $S\cap I$ is measurable in $J$. Then given any two covers you can reduce them to a common subcover.
Yes, that is the result I'm trying to get. How can I show that $S\cap I$ is measurable in $J$. –  Student Feb 8 '13 at 16:28
$S \subset I$ is measurable $\Leftrightarrow I \setminus S$ is measurable $\Leftrightarrow m_\star(S)=m^\star(S) \Leftrightarrow (\forall \epsilon >0)(\exists U \quad open)(\exists K \quad compact)(K \subset S \subset U \land m(U\setminus K)<\epsilon)$ –  Student Feb 8 '13 at 17:29