I am stuck on this problem involving characteristic functions. Say you have two characteristic functions, $\phi_1$ and $\phi_2$, and you are looking at the set $A =\{t : \phi_1(t) = \phi_2(t)\}$. How can you prove (or at least tell) that this set is closed?
Set for which two characteristic functions are equal
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probability
probability-theory
1 Answers
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Since $\phi_1$ and $\phi_2$ are continuous, $\phi_1 - \phi_2$ is continuous, and the set in question is closed because it is the preimage of $\{0\} \subset \mathbb{C}$ under $\phi_1 - \phi_2$. (Recall that a function is continuous if and only if the preimage of every closed set is closed.)
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0oh right, I forgot about that property of characteristic functions. thanks a lot for your help! – 2011-10-11