Measure Theory

[Dhamnekar Win,odd]

The class of finite unions of arbitrary (also unbounded) intervals is an algebra on Ω=R (but is not a σ -algebra). How to prove it? 

I know an algebra is a class of sets A⊂2Ω which holds the following conditions

 

1)Ω∈A

 

2)A is closed under complements.

3)A is closed under unions.

σ -algebra fulfills all these three conditions. But in addition, it is closed under countable unions. Here countable means finite or countably infinite. 

I also know that R=(−∞,+∞) which is uncountably infinite unions of arbitrary (also unbounded) intervals.(is that correct?  🤔) 

Now, with these information available to me,, how can I answer this question? 

Edited Sunday at 07:38 PM by Dhamnekar Win,odd
I added more text.

[Dhamnekar Win,odd]

Errata: an algebra is a class of sets \(\mathcal{A} \subset 2^{\Omega}\)