I’m studying Probability theory, but I can’t fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set of all intervals in [0, 1]. Am I wrong? I just need more examples.
Also I want to understand what is Borel $\sigma$-algebra.
To try and motivate the technical answers, I’m ploughing through this stuff myself, so, people, do correct me:
Imagine Arnold Schwarzenegger‘s height was recorded to infinite precision. Would you prefer to try and guess Arnie’s exact height, or some interval containing it?
But what if there was a website for this game, which provided some pre-defined intervals? That could be quite annoying, if say, the bands offered were $[0,1m)$ and $[1m,\infty)$. I suspect most of us could improve on those.
Wouldn’t it be better to be able to choose an arbitrary interval? That’s what the Borel $\sigma$-algebra offers: a choice of all the possible intervals you might need or want.
It would make for a seriously (infinitely) long drop down menu, but it’s conceptually equivalent: all the members are predefined. But you still get the convenience of choosing an arbitrary interval.
The Borel sets just function as the building blocks for the menu that is the Borel $\sigma$-algebra.