Hello,
denotes the Borel sigma-algebra over the real line.
For any , let's denote the application
1. Show that is a sigma-algebra over the real line.
2. Show that contains all the intervals
3. Show that
Okay, so the main problem is that I don't know how and where to start... my first idea was to start by what's asked in question 3., so I believe this wasn't the right idea lol...
Thanks for any help
Yep, I agree with that, but I can't see how it is possible to define , since there is no closed form for a Borelian ? :O
So in your opinion, is this what they consider as "all the intervals" ? (there is no further detail)For 2., .
Did they intend to say open intervals ?
Okay, so I believe we'll have to use the inclusion between sigma-algebras (argh there must be an ascii code for sigma lol)3. will follow from 1. and 2., because the intervals generate as a σ-algebra.
hehe ^^' thanks ![Linguistic note. The English word for application is mapping, or map.]
You don't need to have a closed form for Borel sets. All you need to do is to show that the collection of all sets of the form (where A is Borel) satisfies the axioms for a σ-algebra. So for example you must check that given a countable collection of sets , their union is also in the range of the mapping .
The same argument will apply to any type of interval (open, closed, unbounded, whatever). The point is that the translate of an interval is an interval.
It is &#863;. There is a full list of html character entity references here. But for some reason, this Forum will only accept the numerical version of the character description, not the name. So &#863; will produce σ, but σ will not.