I'm being introduced to $\sigma$-algebra's and I came across the following definition:

Let $\mathcal{B}(\overline{\mathbb{R}})$ be the $\sigma$-algebra generated by the sets $\{-\infty\}$,$\{\infty\}$ and $B\in\mathcal{B}(\mathbb{R})$. This $\sigma$-algebra $\mathcal{B}(\overline{\mathbb{R}})$ will be called the Borel algebra of $\overline{\mathbb{R}}$.

This definition got me confused as I've never heard of the sets $\{-\infty\}$ and $\{\infty\}$. I always thought that infinity wasn't a number.

Question: What should one think of or imagine when trying to understand the meaning of the (singleton?) $\{\infty\}$?

I know that something similar has been asked here, but I think my question goes a little bit further into what $\{\infty\}$ actually means.

Edit: I want to give an example that explains my confusion. In school I learned that $[a,\infty] = [a,\infty)$.

Thanks in advance!

  • 1
    $\begingroup$ Look back before that for the discussion of $\overline{\mathbb R}$. You will have to understand that before you can understand $\mathcal B(\overline{\mathbb R})$. $\endgroup$ – GEdgar Dec 1 '17 at 10:50
  • $\begingroup$ @GEdgar $\overline{\mathbb{R}}$ is never mentioned! $\endgroup$ – Mr. President Dec 1 '17 at 11:01
  • 1
    $\begingroup$ Either this is a very bad book, or it has some prerequisite where $\overline{\mathbb R}$ is discussed... Otherwise: how can you expect to define $\mathcal B(\overline{\mathbb R})$ without knowing what is $\overline{\mathbb R}$ ? $\endgroup$ – GEdgar Dec 1 '17 at 11:08

The extended real line is $\Bbb R$ adjoined by the two non-number-objects $+\infty$ (or $\infty$) and $-\infty$, along with the declaration that $-\infty<r<\infty$ for any real number $r$.

The text you quote simply describes the Borel sets of the extended real line, and since it turns out that these are exactly the same as adding the singletons $\{-\infty\}$ and $\{\infty\}$ to the Borel sets of the reals, this can be given as a definition instead of meddling with the topology.

Note that in the extended real line, since $\infty$ is now an actual object, we have that $[a,\infty]\neq[a,\infty)$.

(If all this is being confusing, think about $\Bbb R$ as $(0,1)$ and the extended real line as $[0,1]$.)

So what is the meaning of $\{\infty\}$? It's a set, with a single element, and that element is $\infty$. Oh, yes, not only real numbers can be elements of sets. Any mathematical object can be an element of a set.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.