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

Gold Member
MHB
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

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.

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Math Amateur
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.

Last edited:
Math Amateur
Gold Member
MHB
Thanks to Dragon27 and Math_QED for clarifying the issue ...