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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!

Mr. President
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    Look back before that for the discussion of $\overline{\mathbb R}$. You will have to understand that before you can understand $\mathcal B(\overline{\mathbb R})$. – GEdgar Dec 01 '17 at 10:50
  • @GEdgar $\overline{\mathbb{R}}$ is never mentioned! – Mr. President Dec 01 '17 at 11:01
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    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 $\overline{\mathbb R}$ ? – GEdgar Dec 01 '17 at 11:08

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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.

Asaf Karagila
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