Hi there,
I have a question on real analysis, I will write my try and could you please help me to solve it.
we need to prove that f(x)=X^2 ,Q(x)= tanx, and J(x)= -sqrt(x) are borel measurable by the formal definition {f<= b}is in Borel, for all b in R.
Thanks,
What do you know about measure theory ?
Any function f is Borel measurable if (with your definition) for any b in R, is a Borel set.
In a more general way, since is a closed set, it is sufficient for f to be continuous. Thus the pre-image of this set will be a Borel set, since the pre-image of a closed set under a continuous function is a closed set. (and a closed set is a Borel set)
Hi there,
I have a question on real analysis, I will write my try and could you please help me to solve it.
we need to prove that f(x)=X^2 ,Q(x)= tanx, and J(x)= -sqrt(x) are borel measurable by the formal definition {f<= b}is in Borel, for all b in R.
Thanks,
Fix
So as you pointed out, you need to see that is a borel set.
We have that so
There are 3 cases:
If then
If then
If then
The first and last case obviously are borel sets (the empty set is in every -algebra, closed sets are borelian as well (their complement is open) ), what about the second one?
Well, in with the usual metric, singletons are closed, so is closed, therefore is borelian.
Hence for every , is in is measurable.
You have to do this for every of your functions. Also as Moo already told you, if you already know (i bet you do) that the borel -algebra is the algebra generated by all sets of the form :
open (a,b)
OR
closed [a,b]
OR
psemi-open (a,b]
OR
psemi closed [a,b)
(you can even use the above for rational end points still get the same sigma algebra)
So, if \mathbb R,\mathcal B(\mathbb R) ) \longrightarrow (\mathbb R,\mathcal B(\mathbb R) )" alt=" h\mathbb R,\mathcal B(\mathbb R) ) \longrightarrow (\mathbb R,\mathcal B(\mathbb R) )" /> and you show that you have is like any of the above type of sets, FOR EVERY K of any of the above, then h is measurable.