Borel fields and borel sets

Do you understand how the problem of assigning probabilities to possibilities is related to the problem of assigning a "size" to each member of a set of subsets of a set? Do you understand why the domain of the function that assigns sizes must be a sigma algebra? (Some authors use the term "sigma field", but they seem to mean the same thing). Do you understand that the set of Borel sets is defined to be a sigma algebra?

I can explain all of the above, but I don't want to do it if you already understand those things. Maybe you're just trying to find out why the sigma algebra of Borel sets is chosen over all other sigma algebras, like the sigma algebra of Lebesgue measurable sets. I would like actually like to know that too, so I hope someone else will explain that.

 

The Borel sigma-algebra over the real line (actually, you mention a Borel Field, which is something a little different but, as you mention probability, I'll assume, for now, that you meant a sigma-algebra) is the smallest sigma-algebra that contains the intervals (more technically, it contains the open sets); being a sigma-algebra means that it's also closed for countable unions, countable intersections and set complementation.

Why do we need these for probability? Because, when you define a probability measure, you must be able to measure the probability for more compex sets than the intervals; for example, the probability measure is countably addictive so, if the probability of each set in an infinite (disjoint) family is defined, so must the probability of their union. The same goes for the other closure conditions.

The Borel sigma-algebra is the "natural" set algebra over topological spaces; remember that it's defined as the one generated by the open (or closed) sets of the space, and this only makes sense when you have a topology. On the other hand, you dont need to have any topology when defining the (for example) Lebesgue measure. Another feature of the Borel algebra, is that its elements may be explicitly described by (tranfinitely) iterating the so-called [itex]\delta\sigma[/itex]: start with the open sets, form all the countable intersections, then all the countable unions, and repeat until you get to the first uncountable ordinal. This explicit description is very useful in certain branches of Set Theory.

One problem with the Borel algebra is that it's not complete: this means that certain sets that should have null measure don't belong to it but, when you complete it, you get (in the euclidean case), the Lebesgue measure. Problems with incomplete algebras usually arise when you want to extend the measure to higher dimension; this is another reason why the Lebesgue measure is preferred when doing this.