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

In summary, to prove that the function $\phi(t)=t^{-1}$ is Borel measurable, we need to show that it pulls back Borel sets to Borel sets, i.e. if t^{-1} is a Borel set, then t is also a Borel set. This follows from the definition of a measurable function and the fact that t^{-1} is a Borel set.
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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.
 
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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
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?
 

1. What does it mean for a function to be Borel measurable?

Borel measurability is a property of real-valued functions defined on a measurable space. It means that the pre-image of any Borel set under the function is also a measurable set in the domain space. In simpler terms, a Borel measurable function is one that preserves measurability of sets.

2. How is Borel measurability related to the Borel sigma-algebra?

The Borel sigma-algebra is the smallest sigma-algebra that contains all the open sets in a topological space. A function is Borel measurable if and only if its pre-image preserves the Borel sigma-algebra, meaning that the pre-image of any Borel set is also a Borel set.

3. Can you give an example of a function that is not Borel measurable?

Yes, a common example is the characteristic function of the Vitali set, which is a non-measurable set in the real line. This function is not Borel measurable because its pre-image does not preserve the Borel sigma-algebra.

4. What is the significance of Borel measurable functions in mathematics?

Borel measurable functions play a crucial role in the study of measure and integration theory, as well as in probability and statistics. They are also important in the construction of Lebesgue measure and Lebesgue integral, which are fundamental concepts in modern analysis.

5. Are all continuous functions Borel measurable?

No, not all continuous functions are Borel measurable. A function can be continuous but not Borel measurable if its pre-image does not preserve the Borel sigma-algebra. However, all continuous functions on a compact metric space are Borel measurable.

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