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Homework Help: Function is Borel measurable

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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

    3. 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.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 16, 2008 #2


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    Homework Helper

    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.)
  4. Oct 17, 2008 #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?
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