I'm trying to get the central concepts correct, so I'm going to express them without embellishment.

  1. A Borel $\sigma$ algebra is defined as a sigma algebra generated by a topological space $(M,\mathcal{T})$. If $M$ is $\mathbb{R}$, it will typically be the "standard topology" defined as the set of all open intervals.
  2. Because a $\sigma$ algebra is closed under intersection of infinitely-countably many sub-sets, we can prove that $\displaystyle\bigcap_{n\geq1}(a -\frac{1}{n},\,b)= [a,b)$. This step would not be possible within the topological space where only finite intersections are allowed. So we "recover" closed and half-closed intervals, thanks to the Borel $\sigma$-algebra.
  3. Now, the half-open intervals are in the sigma algebra (and so are closed intervals), and can be assigned a measure.

So we went through quite a bit of maneuvering (assuming all this is correct) for what purpose? In particular, if we focus on probability spaces, the probability of a close interval is the same as an open interval, since the probability of any point in the real line is zero.

Here is a tentative, unsophisticated, and crass way of looking at it (I'm posting it here to get further feedback):

We can't assign a probability measure to $[0,1]$ because of Cantor's different types of infinites, and $2^Ω$ of $[0,1]$ being far too big, etc. But we want a smooth domain, so that we can get a nice $pdf$ to integrate over... Solution: start with the standard topology over the real line, and then "fill" the gaps by turning it into a sigma algebra.

Reference: On minute:sec 57:39 of this video the speaker ends up the segment with the words: "That very often is very useful." This prompted my question. Where is it needed in probability?

  • $\begingroup$ There are a lot more sets than just intervals and they can get a bit bizarre, e.g. the Cantor set. $\endgroup$ – Cameron Williams Apr 18 '16 at 23:05
  • $\begingroup$ Yes, I'm familiar with the Cantor set, but I'm just scratching the surface of measure theory and topology, and I'm looking for a motivating statement to justify the "trouble" of these mathematical "acrobatics" in the context of probability spaces. $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:07
  • $\begingroup$ The reason we go through the trouble is that many spaces of interest (in practice) come equipped with a nice topology. The open sets tell us a lot about continuity, so we want to use those to develop the measure space (so that continuous functions are naturally packaged into the theory - we like to integrate continuous functions!). The basic axioms of measure theory tacked on to the topological space is what causes the issues and makes measure theory very complicated in some ways. $\endgroup$ – Cameron Williams Apr 18 '16 at 23:10
  • $\begingroup$ So could you draw the tentative conclusion: "We can't assign a probability measure to [0,1] because of Cantor's different types of infinites, and $2^\Omega$ of [0,1] is far too big, etc. But we want a smooth domain, so that we can get a nice pdf to integrate over... Solution: start with the standard topology over the real line, and then "fill" the gaps by turning it into a sigma algebra. Is that it? $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:16
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    $\begingroup$ I think that's a good way of thinking about it, yes. $\endgroup$ – Cameron Williams Apr 18 '16 at 23:17

That's a very good question. The answer is not so simple, but I would say that there are two main reasons why we like a lot Borel measures on the line (or preferably their Lebesgue counterpart):

  • they are regular, which is a wonderful property to establish many properties of the measures, such as when dealing with the relation with topological properties;

  • they allow defining conditional measures, which is an important concept in further developments of probability theory, such as stochastic processes.

It should be emphasized that both properties (somehow respectively of semilocal and local nature) fail in general for arbitrary Borel measures on arbitrary metrizable spaces.

There other also less simple reasons why we like measures of the type that you mention. These include stisfying a Lebesgue differentiation theorem (not only for the Lebesgue measure, but for the completion of any Borel measure on the line) and in fact being all that is needed to consider Borel measures on $\mathbb R^n$ (which turn out to be superfluous). But this would be a long story.

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  • $\begingroup$ Thanks. Can you please comment on my added appendix to the OP? $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:22
  • $\begingroup$ Your description is really not a good way of looking at measure theory in general, although it is a good starting point for the beginning of the theory. In further developments the canon is completely different from what you describe (say for ergodic theory, even for smooth ergodic theory), the reason being that many of the natural structures are rarely smooth or even continuous! $\endgroup$ – John B Apr 18 '16 at 23:26
  • $\begingroup$ Incidentally, "regularity" (see my answer) is the ultimate property leading to your comments on open/closed sets and their generalizations. $\endgroup$ – John B Apr 18 '16 at 23:32
  • $\begingroup$ I don't understand your last comment. Would it be possible to break apart my comment at the end of the OP for whatever truth it may contain, and also for its clear shortcomings? $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:37
  • $\begingroup$ Well, we certainly don't need a "smooth domain" neither "a nice pdf to integrate", so indeed, it is not quite appropriate to say that we want these. :) $\endgroup$ – John B Apr 18 '16 at 23:39

Defining a measure is defining a rule to measure things. The first thing you do in order to do that is to decide what sets you want to measure: picking a sigma-algebra is precisely the act of deciding what sets you will measure.

Now picking the Borel sigma-algebra is exactly saying that you will measure open sets.

There are no acrobatics involved...

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  • $\begingroup$ "Acrobatics" wasn't meant to elicit controversy, just emphasize the issue in the question. That being said, it just seems as though not happy with open sets, we end up closing them, thanks to the fact that within a sigma algebra we can perform unlimited number of intersections (as opposed to a topology). So what is the motivation behind this? $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:26
  • $\begingroup$ Well, if you intend to measure a set you automatically measure it's complement. $\endgroup$ – Mariano Suárez-Álvarez Apr 18 '16 at 23:27
  • $\begingroup$ As for countable unions, well, experience tells us that we also want that. If you wait a bit, you will see how useful it is to have that available. $\endgroup$ – Mariano Suárez-Álvarez Apr 18 '16 at 23:29
  • $\begingroup$ Can I conclude that if we didn't make sure to include closed and half-closed sets, we wouldn't be able to get the probability of the complement? $\endgroup$ – Antoni Parellada Apr 18 '16 at 23:55
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    $\begingroup$ The motivation is simply that you want to measure open sets. $\endgroup$ – Mariano Suárez-Álvarez Apr 19 '16 at 0:29

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