1. ## sigma-algebra problems

Hello,

$\displaystyle \mathcal{B}(\mathbb{R})$ denotes the Borel sigma-algebra over the real line.
For any $\displaystyle x\in\mathbb{R}$, let's denote $\displaystyle T_x$ the application $\displaystyle T_x(y)=x+y ~,~ y\in\mathbb{R}$

1. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ is a sigma-algebra over the real line.

2. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ contains all the intervals

3. Show that $\displaystyle \forall x\in\mathbb{R} ~,~ T_x(\mathcal{B}(\mathbb{R}))=\mathcal{B}(\mathbb{R })$

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

2. Originally Posted by Moo
$\displaystyle \mathcal{B}(\mathbb{R})$ denotes the Borel sigma-algebra over the real line.
For any $\displaystyle x\in\mathbb{R}$, let's denote $\displaystyle T_x$ the application $\displaystyle T_x(y)=x+y ~,~ y\in\mathbb{R}$

1. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ is a sigma-algebra over the real line.

2. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ contains all the intervals

3. Show that $\displaystyle \forall x\in\mathbb{R} ~,~ T_x(\mathcal{B}(\mathbb{R}))=\mathcal{B}(\mathbb{R })$
For 1., you need to say things like $\displaystyle \textstyle\bigcup_{n=1}^\infty T_x(A_n) = T_x\left(\bigcup_{n=1}^\infty A_n\right)$.

For 2., $\displaystyle [a,b] = T_x([a-x,b-x])$.

3. will follow from 1. and 2., because the intervals generate $\displaystyle \mathcal{B}(\mathbb{R})$ as a σ-algebra.

[Linguistic note. The English word for application is mapping, or map.]

3. Originally Posted by Opalg
For 1., you need to say things like $\displaystyle \textstyle\bigcup_{n=1}^\infty T_x(A_n) = T_x\left(\bigcup_{n=1}^\infty A_n\right)$.
Yep, I agree with that, but I can't see how it is possible to define $\displaystyle A_n$, since there is no closed form for a Borelian ? :O

For 2., $\displaystyle [a,b] = T_x([a-x,b-x])$.
So in your opinion, is this what they consider as "all the intervals" ? (there is no further detail)
Did they intend to say open intervals ?

3. will follow from 1. and 2., because the intervals generate $\displaystyle \mathcal{B}(\mathbb{R})$ as a σ-algebra.
Okay, so I believe we'll have to use the inclusion between sigma-algebras (argh there must be an ascii code for sigma lol)

[Linguistic note. The English word for application is mapping, or map.]
hehe ^^' thanks !

4. Originally Posted by Moo
Originally Posted by Opalg
For 1., you need to say things like $\displaystyle \textstyle\bigcup_{n=1}^\infty T_x(A_n) = T_x\left(\bigcup_{n=1}^\infty A_n\right)$.
Yep, I agree with that, but I can't see how it is possible to define $\displaystyle A_n$, since there is no closed form for a Borelian ?
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 $\displaystyle T_x(A)$ (where A is Borel) satisfies the axioms for a σ-algebra. So for example you must check that given a countable collection of sets $\displaystyle T_x(A_n)$, their union is also in the range of the mapping $\displaystyle T_x$.

Originally Posted by Moo
Originally Posted by Opalg
For 2., $\displaystyle [a,b] = T_x([a-x,b-x])$.
So in your opinion, is this what they consider as "all the intervals" ? (there is no further detail)
Did they intend to say open intervals ?
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.

Originally Posted by Moo
argh there must be an ascii code for sigma lol
It is &#38;&#35;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 &#38;&#35;863; will produce σ, but &sigma; will not.