Is this union of intervals is a borel set?

Question

I have to show that $$ \bigcup_{n=1}^\infty \left[n,n+\frac{1}{n^2}\right]$$ and $$ \bigcup_{n=1}^\infty \left[n,n+\frac{4}{n^2}\right]$$ both are Borel sets.

I thought that unions of closed intervals are always Borel. Am i missing something?

Answer 1

$$\text{open sets are Borel}\stackrel{1}{\to}\text{closed intervals are Borel}\stackrel{2}{\to}\text{countable unions of closed intervals are Borel}$$

Collection of Borel sets is $\sigma$-algebra generated by open sets, so:

• 0) open sets are Borel.
• 1) since the collection is closed under complements.
• 2) since the collection is closed under countable unions.

Answer 2

First time answering here so pardon any mistakes in protocol. Here's a start. You can rewrite the closed sets as countable intersections of open sets. Then you could take some additional steps and probably use complements of closed sets to complete the proof. Hopefully my Tex conversion tool works.

$\bigcup\limits_{n=1}^{\infty }{{}}\left[ n,n+\frac{1}{{{n}^{2}}} \right]=\bigcup\limits_{n=1}^{\infty }{\bigcap\limits_{k=1}^{\infty }{\left( n-\frac{1}{k},n+\frac{1}{{{n}^{2}}}+\frac{1}{k} \right)}}$