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

In summary, the conversation is about a proof in Sheldon Axler's book on Measure, Integration & Real Analysis. The specific focus is on Result 2.7 and its proof, which involves taking the infimum of a term over all sequences of open intervals whose union contains a given set. The question is asking for an explanation of why this leads to the inequality ##\mid t + A \mid \leq \mid A \mid##. In response, Peter suggests using a lemma from previous questions to help understand the proof.
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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:
Axler - Result 2.7 - outer measure is translation invariant .png


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
 
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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)##.
 
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Thanks Math_QED ...
 

1. What is translation invariance of outer measure?

Translation invariance of outer measure, also known as shift invariance, is a property of outer measures, which are functions that assign a value to sets in a given space. It states that the outer measure of a set remains unchanged when the set is translated by a fixed amount.

2. How is translation invariance related to Result 2.7 in Axler's book?

Result 2.7 in Axler's book, "Measure Theory for Mathematicians," states that if a set is measurable, then its outer measure is equal to its inner measure. This result is closely related to translation invariance, as it implies that the outer measure of a translated set will be equal to its inner measure, and therefore remain unchanged.

3. Why is translation invariance important in measure theory?

Translation invariance is important in measure theory because it allows us to extend the concept of measure to more general spaces, such as infinite-dimensional spaces. It also allows us to define and work with measures in a more flexible and intuitive way, as we can apply translations to sets without changing their measure.

4. Can you provide an example of translation invariance in action?

One example of translation invariance is the Lebesgue outer measure on the real line. If we have a set A with Lebesgue outer measure m*(A), and we translate A by a distance of 2, we will still have the same outer measure, as m*(A+2) = m*(A).

5. Are there any other important properties of outer measures?

Yes, there are several other important properties of outer measures, including countable subadditivity, monotonicity, and continuity from above and below. These properties, along with translation invariance, help to define and characterize measures and make them useful tools in various branches of mathematics, including analysis, probability, and geometry.

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