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Let $f$ be defined on a measurable set $E$. Show that $f$ is measurable if and only if for every Borel set $A$, $f^{-1}(A)$ is measurable.
Hint: The collection of sets $A$ that have the property that $f^{-1}(A)$ is measurable is a $\sigma$-algebra.

I want to know if the following proof is correct:

$\mathcal{A}$ is the collection in question, $\mathcal{B}$ is the set of Borel sets, and $\mathcal{M}$ is the set of measurable sets.

  • Let $c \in \mathbb{R}$ arbitrarily chosen. Then $(c, \infty)$ is open so it is in $\mathcal{B}$ and $f^{-1}((c,\infty)) \in \mathcal{M}$.
  • Also since $f^{-1}(\{\infty\}) \in \mathcal{M}$ so $\{\infty\} \in \mathcal{A}$.
  • By both of the above this shows that $(c, \infty] \in \mathcal{A}$.
  • Let $a,b \in \mathbb{R}$. Since $f$ is measurable $f^{-1}([-\infty,b))$ and $f^{-1}((a,\infty])$ are both in $\mathcal{M}$ so $[-\infty,b)$, $(a,\infty]$ are both in $\mathcal{A}$ and $f^{-1}([-\infty,b)) \cap f^{-1}((a,\infty])=f^{-1}((a,b)) \in \mathcal{M}$. So $(a,b)\in A$. This shows that all open intervals are in $\mathcal{A}$.
  • For $A \in \mathcal{B}$, $f^{-1}(A) \in \mathcal{M}$ so $A \in \mathcal{A}$.
  • $f^{-1}(\emptyset) \in \mathcal{M}$, so $\emptyset \in \mathcal{A}$.

I've gotten $\LaTeX$ fatigue, but I think all that is left to show is that countable unions and complements are in $\mathcal{A}$ to complete the definition of $\sigma$-algebra.

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Well the 2nd and 3rd step seem a bit unnecessary to me. I had done this in a slightly different way.To put into perspective, the "nice" properties that inverse functions satisfy are enough to do most of the required work. Just look at how you can preserve the complements and unions of sets. Maybe I can try and give an explicit answer later; I don't know whether you want one. Hope this helps. – Vishesh Oct 19 '12 at 4:15
I would like to compare my answer to someone else certainly, especially to see if there is a more efficient/eloquent way instead of my list. – emka Oct 19 '12 at 4:18
What's your definition of measurability and w.r.t. what $\sigma$-algebras? Because from the point of view of the theory of general measurable spaces, your problem is the definition of measurability. [Wikipedia]. – kahen Oct 19 '12 at 5:28
up vote 4 down vote accepted

For your proof, I'm not quite sure I understand where you are going with your list of bullet points.

In the beggining it seems like you are assuming $f$ is measurable, and trying to prove that $f^{-1}(B)$ is measurable for any Borel set $B$, but your second to last bullet claims that $f^{-1}(B)$ is measurable for any borel set $B$, which then makes it seem like you are assuming this and trying to prove measurability. (But if you are assuming this, then all the other bullet points are trivial). Whichever it is you're trying to do you should make clear at the beginning of the proof.

If the first is the case (which I think it probably is) what you want to do is show that the sets form a $\sigma$-algebra which contains the open sets, as this is the reason we can say that $f^{-1}(B)$ is measurable for every Borel set.

Here is a sample proof highlighting what I've said above, since you said in the comments you would want an explicit answer.

($\Leftarrow $) if $f^{-1}(A)$ is measurable for every Borel set, then in particular $f^{-1}(A)$ is measurable for every open set, so $f$ is measurable.

($\Rightarrow$) if $f$ is measurable then we know that $f^{-1}(A)$ is measurable for every open set.

Moreover, Let $\mathcal{A}$ be the collection of sets $A$ satisfying $f^{-1}(A)$ is measurable. In particular we know that $\mathcal{A}$ contains all open sets by above. Then let $A_1, A_2, \cdots \in \mathcal{A}$

Then $f^{-1}(\bigcup_1^{\infty} A_n) = \bigcup_1^{\infty} f^{-1}(A_n)$ is measurable since it is the countable union of measurable sets, so $\bigcup_1^{\infty} A_n \in \mathcal{A}$. Similarly, $f^{-1}(A_1^{c}) = f^{-1}(A_1)^c$ is measurable since it is the compliment of a measurable set, so $A_1^{c} \in\mathcal{A}$. Thus $\mathcal{A}$ is a $\sigma$-algebra, which contain the open sets, so $\mathcal{A}$ contains all borel sets and we're done.

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So I didn't have to explicitly enumerate different types of open intervals? This makes me feel inefficient. – emka Oct 19 '12 at 5:21

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