What is the Borel Hierarchy?

Question

I'm taking a measure theory class, and our professor mentioned in passing that the Borel sets "stratify" into a hierarchy. But I'm getting lost in the $\Pi$s, $\Sigma$s, and $\Delta$s on the wikipedia page.

Would someone mind explaining

1. What the Borel hierarchy is.
2. Why we might find it interesting.

Are there interesting things we can prove about $\Pi_1^0$ sets that we can't prove about $\Pi_2^0$ sets? What about $\Sigma_3^0$ versus $\Sigma_4^0$? etc. I'm willing to believe that theorems like this might exist "low down" the hierarchy, but what about the differences between a $\Pi_\alpha^0$ and a $\Pi_\beta^0$ for $\alpha$ and $\beta$ infinite ordinals? Can we prove things about them too? If not, why study them? I don't mean this to sound offensive, I'm just confused.

Please help! Thanks!

Answer 1

The idea of how these sets come about is pretty natural when you are thinking about algebras of sets in measure theory. The name of the game is: what can I make from open sets using countable unions, countable intersections, and complementation?

See, open sets are stable under finite unions and intersections, so nothing new is found. The same is true for countable unions. But countable intersections? Now that's something new, something we give a name to: $G_\delta$ sets. Dualizing with closed sets, you get $F_\sigma$ sets.

Now, $G_\delta$ sets are stable under countable intersections... but no longer under countable unions! So you're going to have to do something about those too. That's when you get $G_{\delta\sigma}$ sets.

These iterations are what the $\Sigma$ and $\Pi$ notation track. It develops a sort of stratification of types of sets, ones which are accessible via so-and-so many iterations of the process.

Now, this is where my knowledge on the topic runs out, but the wiki page seems pretty informative on why we might find them interesting:

1. "The Borel hierarchy is of particular interest in descriptive set theory."

2. "One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank."

3. "Properties of sets of small finite ranks are important in measure theory and analysis."