What is a Borel Measurable Function and How to Prove It?

[sdf123]

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.

 

[morphism]

To get your TeX to show up, enclose it in (for inline / text style) or (for equation style) tags.<br /> <br /> 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<sup>-1</sup>(E) is in A whenever E is in B.<br /> <br /> 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.)

 

[sdf123]

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?