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

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In summary, the conversation discusses the implications of Sheldon Axler's definition of a ##\sigma##-algebra and its relation to the set of all subsets of ##\mathbb{R}##. It is mentioned that we cannot define Lebesgue measure on this ##\sigma##-algebra, but it is not stated that the set itself is not a ##\sigma##-algebra. The conversation also mentions the desire for the set to satisfy the properties in the definition of a ##\sigma##-algebra and thanks the contributors for their help.
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Math Amateur

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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:

Axler - Sigma Algebres ... Page 26 .png


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
 
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Math Amateur said:
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 said:
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.
 
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Thanks to Dragon27 and Math_QED for clarifying the issue ...

Much appreciate your help ...

Peter
 

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