• Support PF! Buy your school textbooks, materials and every day products Here!

Function is Borel measurable

  • Thread starter sdf123
  • Start date
  • #1
2
0

Homework Statement



Prove that the function $\phi(t)=t^{-1}$ is Borel measurable.

Homework Equations



Any measurable function into $ (\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $ \mathcal{B}(\mathbb{R})$ is the Borel sigma algebra of the real numbers $ \mathbb{R}$, is called a Borel measurable function

The Attempt at a Solution



I think I need to prove that t^{-1} is a Borel set, and so prove that it is open? I am quite unclear on the actual definition of a borel measurable function, and that is perhaps my problem.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
2,015
4
To get your TeX to show up, enclose it in [itex] (for inline / text style) or [tex] (for equation style) tags.

Now, are you familiar with the definition of a measurable function? Say you have two measurable spaces X and Y with sigma-algebras A and B, respectively. A function f:X->Y is (A-B) measurable if it pulls back sets in B to sets in A, i.e. if f-1(E) is in A whenever E is in B.

A Borel measurable function f:X->Y is then an (A-B) measurable function, where B is the Borel sigma-algebra on Y. (Of course for this to make sense, Y has to be a topological space.)
 
  • #3
2
0
So, in order to prove that \phi(t)=t^{-1} is Borel measurable, I need to show that if t^{-1} is a Borel sigma algebra, that {t^{-1}}^-1=t is in t, which it obviously is?
 

Related Threads on Function is Borel measurable

  • Last Post
Replies
0
Views
4K
Replies
0
Views
1K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
0
Views
982
Replies
2
Views
984
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
8
Views
1K
Replies
5
Views
2K
Replies
4
Views
863
  • Last Post
Replies
0
Views
892
Top