MHB Sigma Algebras .... Axler Page 26 ....

[Math Amateur]

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

 

[Opalg]

The set of all subsets of $\Bbb{R}$ is a $\sigma$-algebra (but it is not a useful $\sigma$-algebra in the context of measure theory).

 

[Math Amateur]

The set of all subsets of $\Bbb{R}$ is a $\sigma$-algebra (but it is not a useful $\sigma$-algebra in the context of measure theory).

Thanks for the help Opalg ... but can you help further ...

... can you explain what you mean by your statement that the set of all subsets of $\Bbb{R}$ is not a useful $\sigma$-algebra in the context of measure theory ...

Peter