1.5.29 - Let $E$ be a Lebesgue measurable set.
a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$.
b.) If $m(E) > 0$, then $E$ contains a nonmeasurable set. (It suffices to assume $E\subset [0,1]$. In the notation of section 1.1, $E = \bigcup_{r\in R}E\cap N_r$).
Proof a.) Let $E\subset N$ and $N\subset [0,1)$. In the notation of section 1.1, let $$E_r = \{x + r: x\in E\cap [0,1-r)\}\cup \{x+r-1:x\in E\cap [1-r,1)\}$$ Then for all $r\in R = \mathbb{Q}\cap [0,1)$, since $E_r \subset N_r$ and the collection $\{N_r\}_{r\in R}$ is a disjoint collection, it follows that $\{E_r\}_{r\in R}$ is a disjoint collection. By the translation invariance and finite additivty of Lebesgue measure, we have that each $E_r$ is measurable and $m(E_r) = m(E)$ for all $r\in\mathbb{R}$. So $$1\geq m\left(\bigcup_{r\in R}E_r\right) = \sum_{r\in R}m(E_r) = \sum_{r\in R}m(E)$$ and therefore $m(E) = 0$? Not sure if this is right.
Proof b.) Suppose that $E\subset [0,1]$ is a subset with the property that $E$ is measurable. Then for each $r\in R$, the set $E\cap N_r$ is measurable. By the translation invariance and finite additivity of Lebesgue measure, the set $E_{1-r}\cap N$ is therefore measurable, and hence must have measure $0$ by part a.). Since the collection $\{N_r\}_{r\in R}$ is disjoint we have that $$m(E) = m\left(\bigcup_{r\in R}(E\cap N_r)\right) = \sum_{r\in R}m(E\cap N_r) = \sum_{r\in R}m(E_{1-r}\cap N) = 0$$
Not sure if this is right either. Any suggestions on these is greatly appreciated.
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$\begingroup$You proof of part a.) is correct. I copy it here just to add a smal comment.
On the other hand, your proof of part b.) needs some corrections
1.5.29 - Let $E$ be a Lebesgue measurable set.
a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$.
b.) If $m(E) > 0$, then $E$ contains a nonmeasurable set. (It suffices to assume $E\subset [0,1]$. In the notation of section 1.1, $E = \bigcup_{r\in R}E\cap N_r$).
Proof a.) Let $E\subset N$ and $N\subset [0,1)$. In the notation of section 1.1, let $$E_r = \{x + r: x\in E\cap [0,1-r)\}\cup \{x+r-1:x\in E\cap [1-r,1)\}$$ Then for all $r\in R = \mathbb{Q}\cap [0,1)$, since $E_r \subset N_r$ and the collection $\{N_r\}_{r\in R}$ is a disjoint collection, it follows that $\{E_r\}_{r\in R}$ is a disjoint collection. By the translation invariance and finite additivty of Lebesgue measure, we have that each $E_r$ is measurable and $m(E_r) = m(E)$ for all $r\in\mathbb{R}$. So $$1\geq m\left(\bigcup_{r\in R}E_r\right) = \sum_{r\in R}m(E_r) = \sum_{r\in R}m(E)$$
Since $R$ is countable infinite, we have that $m(E)=0$.
Proof b.) Note that if $F$ is measurable and $F \subset N_r$, for some $r\in R$. Then, let $$F{-r}=\{x-r : x \in F \cap [r,1)\}\cup \{x-r+1 : x \in F \cap [0,r)\}$$We have that $F{-r}$ is measurable, $\mu(F)=\mu(F_{-r})$ and $F{-r} \subset N$. From part a.) we get $\mu(F_{-r})=0$ and so $\mu(F)=0$.
So we have proved that, if $F$ is measurable and $F \subset N_r$, for some $r\in R$, then $\mu(F)=0$
Now, suppose that $E\subset [0,1]$ is a subset with the property that $E$ is measurable and $\mu(E)>0$. Suppose, that for all $r \in R$, $E \cap N_r$ are measurable. Since $E \cap N_r \subset N_r$, we have $\mu(E \cap N_r)=0$.
But, since $[0,1)$ is the disjoint union of $N_r$'s, we have $E\cap [0,1)$ is the disjoint union of $E \cap N_r$'s. So we get
$$ 0< \mu(E)=\mu(E\cap [0,1))=\sum_{r \in R} \mu(E \cap N_r)= 0$$
Contradiction. So, there is $r \in R$, such that $E \cap N_r$ is not measurable. So $E$ contains a non-measurable set.
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