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This seems intuitively obvious, but I'm struggling with showing it explicitly. My attempt is to shatter $\lvert Z\rvert \in B$ over the possible outcomes of $\mathrm{Sgn}(Z)$. It goes something like:

Let $B$ be a Borel set.

$$P(\lvert Z\rvert \in B)= P(\lvert Z\rvert \in B\mid \mathrm{Sgn}(Z)=1)P(\mathrm{Sgn}(Z)=1) + P(\lvert Z\rvert \in B\mid \mathrm{Sgn}(Z)=-1)P(\mathrm{Sgn}(Z)=-1)$$

I know $P(\mathrm{sgn}(Z)=1)=P(\mathrm{sgn}(Z)=-1) = 1/2$ and that combined with the symmetry of $Z$ leads to the result I want, but only for $\mathrm{Sgn}(Z)= \pm 1$. Am I even close with this approach? Thanks in advance.

Math1000
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Muselive
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    See this question: https://math.stackexchange.com/questions/164806/are-x-and-operatornamesgnx-independent?rq=1 – Math1000 Nov 21 '19 at 01:48
  • That's very helpful, thank you. To be sure; it seems that I don't need to check the condition for an arbitrary Borel set, but rather I'm free to use any generating set. Is this correct? – Muselive Nov 21 '19 at 12:48
  • Sure, in particular sets of the form $(-\infty,x]$. – Math1000 Nov 21 '19 at 17:29

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