Sigma Algebras .... Axler, Page 26 ....

[Math Amateur]

TL;DR Summary

I need help in order to fully understand the implications of Axler's definition of a $\sigma$-algebra ... ...

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

I need help in order to fully understand the implications of Axler's definition of a $\sigma$-algebra ... ...

The relevant text reads as follows:

Now in the above text Axler implies that the set of all subsets of $\mathbb{R}$ is not a $\sigma$-algebra ... ...

... BUT ... which of the three bullet points of the definition of a $\sigma$-algebra is violated by the set of all subsets of $\mathbb{R}$ ... and how/why is it violated ...
Help will be much appreciated ...

Peter

 

[member 587159]

Summary:: I need help in order to fully understand the implications of Axler's definition of a $\sigma$-algebra ... ...

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

I need help in order to fully understand the implications of Axler's definition of a $\sigma$-algebra ... ...

The relevant text reads as follows:

View attachment 267217

Now in the above text Axler implies that the set of all subsets of $\mathbb{R}$ is not a $\sigma$-algebra ... ...

... BUT ... which of the three bullet points of the definition of a $\sigma$-algebra is violated by the set of all subsets of $\mathbb{R}$ ... and how/why is it violated ...
Help will be much appreciated ...

Peter

Axler does not say that. The collection of all subsets is a $\sigma$-algebra (trivially). Axler says that we cannot define Lebesgue-measure on this $\sigma$-algebra and that's why we define Lebesgue measure on Borel sets.

 

[Dragon27]

Now in the above text Axler implies that the set of all subsets of $\mathbb{R}$ is not a $\sigma$-algebra

No, he doesn't. The set of all subsets of $\mathbb{R}$ is obviously a $\sigma$-algebra. It's just that we can't extend the notion of length on all the subsets of $\mathbb{R}$ without violating some highly desirable properties we want it to have (like countable additivity). So we force ourselves to give up on the idea of using all the subsets of $\mathbb{R}$ as the domain of our measure, but we still want this set of subsets to satisfy the properties in the definition of $\sigma$-algebra.

 

[Math Amateur]

Thanks to Dragon27 and Math_QED for clarifying the issue ...

Much appreciate your help ...

Peter