Translation Invariance of Outer Measure .... Axler, Result 2.7 ....

[Math Amateur]

TL;DR

I need help in order to fully understand Axler's proof of the translation invariance of outer measure ...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need help with the proof of Result 2.7 ...

Result 2.7 and its proof read as follows:

In the above proof by Axler we read the following:

" ... ... Thus

... $\mid t + A \mid \leq \sum_{ k = 1 }^{ \infty } l ( t + I_k ) = \sum_{ k = 1 }^{ \infty } l ( I_k )$

Taking the infimum of the last term over all sequences $I_1, I_2, ...$ of open intervals whose union contains $A$, we have $\mid t + A \mid \leq \mid A \mid$. ... ..."Can someone please explain exactly how/why taking the infimum of the last term over all sequences $I_1, I_2, ...$ of open intervals whose union contains $A$, we have $\mid t + A \mid \leq \mid A \mid$ ... ?...Peter

 

[member 587159]

Use a similar lemma as in your previous questions:

If $A$ is a non-emptyset of $\Bbb{R}$ and $b$ is a real number with $b\leq a$ for all $a\in A$, then $b\leq \inf(A)$.

 

[Math Amateur]

Thanks Math_QED ...