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

In summary, the conversation discusses Result 2.7 from Sheldon Axler's book on Measure, Integration & Real Analysis and its proof. The proof shows that by taking the infimum of the last term over all sequences of open intervals whose union contains A, we have $\mid t + A \mid \leq \mid A \mid$. This is due to the general principle that "weak inequalities are preserved by limits" and the fact that the infimum is the greatest lower bound of the set. The conversation also acknowledges the helpfulness of Opalg's reply in explaining this concept.
  • #1
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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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  • #2
The inequality $|t+A| \leqslant \sum_{k=1}^\infty l(I_k)$ shows that $|t+A|$ is a lower bound for the set of sums of the form $\sum_{k=1}^\infty l(I_k)$. The inf of that set is by definition the greatest lower bound of the set. So any other lower bound, in particular $|t+A|$, is less than or equal to that inf.

Peter, it seems to me that most of your recent questions have been, in one form or another, instances of the general principle that "weak inequalities are preserved by limits": if every member of a set satisfies a weak inequality then the limit (or as in this case the sup or inf) of the set satisfies the inequality.
 
  • #3
Thanks for a most helpful reply, Opalg ...

I think you are correct ...but then I am not used to thinking of sup and inf in terms of limits ...

Peter
 

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

1. What is translation invariance of outer measure in mathematics?

Translation invariance of outer measure is a concept in measure theory that states that the measure of a translated set is equal to the measure of the original set. In other words, shifting a set by a fixed amount does not change its measure. This property is important in many areas of mathematics, including analysis and probability theory.

2. What is Result 2.7 in Axler's textbook?

Result 2.7 in Axler's textbook "Measure, Integration & Real Analysis" states that the outer measure of a countable union of sets is less than or equal to the sum of the outer measures of the individual sets. This result is used to prove the translation invariance of outer measure.

3. How is translation invariance of outer measure related to Lebesgue measure?

Lebesgue measure is a specific type of outer measure that is translation invariant. This means that the measure of a translated set is equal to the measure of the original set. In fact, the Lebesgue measure is the unique translation invariant outer measure on the real line.

4. Why is translation invariance of outer measure important in mathematics?

Translation invariance of outer measure is important because it allows for the development of rigorous mathematical theories, such as Lebesgue integration and probability theory. It also provides a way to measure sets that may not have a well-defined volume or area, such as fractals.

5. Can translation invariance of outer measure be extended to higher dimensions?

Yes, translation invariance of outer measure can be extended to higher dimensions. In fact, the concept of translation invariance is applicable to any space with a translation operation defined. This includes vector spaces, metric spaces, and even abstract spaces such as topological spaces.

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