Understanding the proof of Proposition 10 in Ch.2 in Real analysis by Royden and Fitzpatrick "Fourth Edition "

Question

Here is the proposition and its proof:

My question is:

I do not understand how the last equality came from the one just before it, could anyone explains this for me, please?

Answer 1

It’s just invariance of $m^*$ under translation, in this case by $y$: $x\in[A-y]\cap E$ iff $x+y\in A\cap[E+y]$, and $x\in[A-y]\cap E^C$ iff $x+y\in A\cap[E+y]^C$. That is,

$$A\cap[E+y]=\big([A-y]\cap E\big)+y\;,$$

and

$$A\cap[E+y]^C=\left([A-y]\cap E^C\right)+y\;.$$

Answer 2

Observe that we have the equality of sets: $$ y + ([A - y] \cap E) = A \cap [E + y] \quad \text{and} \quad y + ([A - y] \cap E^C) = A \cap [E + y]^C. $$ This can be seen just from the definitions; for example, $$ \begin{align} &x + y \in y + ([A - y] \cap E)\\ \iff\quad &x \in A - y \text{ and } x \in E\\ \iff\quad &x + y \in A \text{ and } x \in E\\ \iff\quad &x + y \in A \text{ and } x + y \in E + y\\ \iff\quad &x + y \in A \cap [E + y]. \end{align} $$ Hence, $y + ([A - y] \cap E) = A \cap [E + y]$. Similarly, for the other equality.

Now, since the outer measure $m^*$ is translation invariant, we have that $$ m^*(A \cap [E + y]) = m^*(y + ([A - y] \cap E)) = m^*([A - y] \cap E) $$ and $$ m^*(A \cap [E + y]^C) = m^*(y + ([A - y] \cap E^C)) = m^*([A - y] \cap E^C). $$