0

A Borel set of $\mathbb R$ is equivalent to an interval. An element of $\mathscr P(\mathbb R)$ is also an interval.

So, can we say a Borel set on $\mathbb R$ is a part of $\mathbb R$?

More generally, what is the use of Borel sets of a set $X$ if we already have $\mathscr P(X)$ there for us?

PS: I am not very confident with Borel sets. What I am stating above might be wrong...

niobium
  • 1,177
  • 1
    There are some strange elements of $\mathscr P(\mathbb R)$, but there are some not-strange elements that are not intervals. For instance, the union of two intervals need not be an interval. – 311411 Oct 02 '22 at 13:26
  • 2
    In what way is a Borel set equivalent to an interval? For example, the set of rational numbers and the set of irrational numbers are particularly simple Borel sets -- how are either of them equivalent to an interval? And there are plenty of elements of $\mathscr P(\mathbb R)$ that are not intervals. Regarding what is the use of Borel sets of a set $X$ if we already have $\mathscr P(X)$ there for us, you may as well similarly ask "what is the use of the even integers in $\mathbb R$ if we already have all numbers in $\mathbb R$ there for us". – Dave L. Renfro Oct 02 '22 at 13:27
  • I am sorry, I misused the word "interval".. – niobium Oct 02 '22 at 13:54

1 Answers1

1

A Borel set is absolutely a part of $\mathbb{R}$ as it is generated by the open sets (as well as the closed and compact sets) on $\mathbb{R}$. Any set that you can imagine or write down is pretty much guaranteed to be a Borel Set. Singletons? Borel. Closed sets? Borel. Half open to infinity? Borel.

Why do we need Borel sets and not just the power set? It is due to the existence of some strange sets on $\mathcal{P}(\mathbb{R})$ that aren't measureable/can't be assigned a meaningful length, although concretely writing such a set on a piece of paper is impossible. But they do exist. For further reading on this matter, check out Vitali's set.

Fid
  • 26