The definition in my book is:
The collection $B$ of Borel sets of real numbers is the smallest
$\sigma$-algebra of sets of real numbers that contains all of the open
sets of real numbers.
First, I'm a bit confused by the wording here.
Does this mean that if a collection $B$ is a $\sigma$-algebra and it contains all of the open sets of real numbers, then the sets in $B$ are called Borel sets?
I'm reading through some of the other answers to this question now, but it seems like different texts use different definitions, so if anyone could shed some light on the definition I provided, that would be helpful.
Best Answer
The definition you gave is the standard one: the smallest $\sigma$-algebra containing all of the open sets. If you know what a $\sigma$-algebra is, and if you know that the intersection of $\sigma$-algebra on a given set is always a $\sigma$-algebra, then this defines the Borel sets, but it does not really tell you how to find all of them, or even how to find any set that is not a Borel set. The definition is a bit tricky (but very useful).
Obviously, every open set is a Borel set. In a $\sigma$-algebra you can take countable intersections, so any countable intersection of open sets is a Borel set. Now you can take unions of such, and these are again Borel sets. This goes on forever, taking countable intersections of such, and unions, and intersections, etc. You can also start by noticing that since every $\sigma$-algebra is closed under complements, all the closed sets are Borel sets. Countable unions of such are also Borel. Countable intersections of such are also Borel, and so on and so on. This can be taken to the transfinite level of repeatedly taking countable unions of intersections of unions of intersections .... to get ever more and more Borel sets, and that will not exhaust all of them.
This gives you a sense of what Borel sets: Potentially extremely complicated sets. The fact that not all sets are Borel sets is well-known, but not a triviality.
In light of the description above, the slick definition is rather impressive. Even if it does leave you know really knowing which set is Borel and which is not, you can still accomplish quite a lot. Most importantly: do not try to wrap your head around each and every Borel set. It suffices to wrap your head around the concept of the Borel sets.