Let $A$ be a subset of $\mathbb{R}$ and $c \in \mathbb{R}$. Define $A+c$ to be the set $$ A+c=\{x+c \mid x \in A\} $$ a) Prove $m^{*}(A+c)=m^{*}(A)$.
b) Prove that $A+c$ is Lebesgue measurable if and only if $A$ is Lebesgue measurable.
a) We know that, $$ m^{*}(A)=\inf \left\{\sum_{k=1}^{\infty} \ell\left(I_{k}\right):\left(I_{k}\right)_{k \in \mathbb{N}} \text { is a sequence of open intervals with } A \subset \bigcup_{k=1}^{\infty} I_{k}\right\} $$
Here, $A \subset \bigcup_{k=1}^{\infty} I_{k}$ then $A+c \subset \bigcup_{k=1}^{\infty} I_{k}+c$
And we know that, $\ell\left(I_{k}\right)=\ell\left(I_{k}+c\right)$ then $m^*(A+c)=\sum_{k=1}^{\infty} \ell\left(I_{k}+c\right)=\sum_{k=1}^{\infty} \ell\left(I_{k}\right)=m^*(A)$
b) A set $E \subseteq \mathbb{R}^{n}$ is Lebesgue measurable if for every $\epsilon>0$ there is an open set $G$ so that $E \subseteq G$ and $$ m^{*}(G \backslash E)<\epsilon \text {. } $$
$(\Longleftarrow)$ Hence, for any $\epsilon>0\:\exists G_{\epsilon}:A\subset G_{\epsilon}\: \land \: m^*(G_{\epsilon}\setminus A)<\epsilon $
Now, $A\subset G_{\epsilon} \stackrel{?}{\implies} A+c\subset G_{\epsilon}+c$ and $\ell(A+c)=\ell(A),\: \ell(G_{\epsilon}+c)=\ell(G_{\epsilon})$
Then can I say $$m^*((G_{\epsilon}+c)\setminus (A+c)) \stackrel{?}{=} m^*(G_{\epsilon}\setminus A)<\epsilon$$
$(\implies)$ We just need to take $-c$ and argue the same thing.
It will be a great help if anyone verify my proof. And suggest me any kind of improvement.
I would write \begin{eqnarray} m^{*}(A) &=& \inf \left\{\sum_{k=1}^{\infty} \ell\left(I_{k}\right) \mid I_{k}\text { open intervals, } A \subset \bigcup_{k=1}^{\infty} I_{k}\right\} \\ &=& \inf \left\{\sum_{k=1}^{\infty} \ell\left(I_{k} +c\right) \mid I_{k}\text { open intervals, } A \subset \bigcup_{k=1}^{\infty} I_{k}\right\} \\ &=& \inf \left\{\sum_{k=1}^{\infty} \ell\left(I_{k} +c\right) \mid I_{k}+c\text { open intervals, } A+c \subset \bigcup_{k=1}^{\infty} I_{k}+c\right\} \\ &=& \inf \left\{\sum_{k=1}^{\infty} \ell\left(J_{k}\right) \mid J_{k}\text { open intervals, } A+c \subset \bigcup_{k=1}^{\infty} J_{k}\right\} \\ &=& m^*(A+c) \end{eqnarray}
For Part (b) I would note that $(U \setminus A) +c = (U +c) \setminus (A +c)$, so $m^* (U \setminus A) < \epsilon$ iff $m^*((U +c) \setminus (A +c) ) < \epsilon$.