Question about particular proof of the translation invariance of the Lebesgue outer measure.

Question

I have a question about the proof of the translation invariance of the Lebesgue outer measure that appears here: http://mathonline.wikidot.com/translation-invariance-of-the-lebesgue-outer-measure

I don't understand how equation (*) follows. I also don't understand how equation (**) follows, but if I figured out the first one, I think I would understand it.

I agree with equation (1) that $m^*(E+a) \leq \sum_{n=1}^\infty l(I_n)$. I also think that from the definition of the outer measure, we have that $m^*(E) \leq \sum_{n=1}^\infty l(I_n)$. But how are they concluding that $$ m^*(E+a) \leq m^*(E)$$ from this information? I think I'm missing something simple, but I've spent several hours thinking about it and can't work it out exactly.

Answer 1

This is a general property of $[0,\infty]$.

If $x \leq a$ for all $a \in A$, then $x \leq \inf A$.

This can be seen by contrapositive. If $x \geq \mathrm{inf A}$, then there exists some $a \in A$ so that $x \geq a$, else $x$ would be a lower bound which contradicts the definition of $\mathrm{inf} A$.

For measurable sets, we can restrict the outer measure which is the inf of the sum of lengths for sequences of open covers.

Answer 2

If (*) would not be true, or equivalently if $m^*(E)<m^*(E+a)$ then covering intervals $I_n$ must exist with: $$m^*(E)\leq\sum_{n=1}^{\infty}l(I_n)<m^*(E+a)$$ contradicting the former bullet.