can anyone intuitively explain me what does a borel field and a borel set mean?Why do we need a Borel field to define all our definitions in probability?
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.can anyone intuitively explain me what does a borel field and a borel set mean?Why do we need a Borel field to define all our definitions in probability?
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.I would like actually like to know that too, so I hope someone else will explain that.