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I am studying probability theory and want to solve the following exercise.

Exercise. Let $X$ be a real random variable. Then the following are equal:

  • $\mathbb{P}\left( X \in a + b\mathbb{Z}\right) = 1$ for some $a, b \in \mathbb{R}$.
  • There exists a $z \in \mathbb{R}$ such that $\lvert \mathbb{\hat{P}}\left(z\right)\rvert = 1$.

Here, $\mathbb{\hat{P}}$ is the characteristic function of the probability measure $\mathbb{P}$.

I was able to show "$\Rightarrow$" but for the other direction I guess I need this statement:

For any probability measure $\mathbb{P}$, if we have

$$ \left\lvert \int_\mathbb{R} e^{itx} d\mathbb{P}\left(x\right) \right\rvert = 1 \;\text{for a}\; t \in\mathbb{R}\setminus\{0\}, $$

then $\exp(itx) = u$ for a $u \in \mathbb{C}$ with $\lvert u \rvert = 1$ (almost surely).

This is clear to me from graphical considerations but I have no clue how to show it formally.

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  • $\begingroup$ is the first bullet supposed to be "for some $a,b \in \mathbb{R}$"? $\endgroup$ – mathworker21 Nov 29 '18 at 17:43
  • $\begingroup$ @mathworker21Yes. I'll change that. $\endgroup$ – fpmoo Nov 29 '18 at 17:45
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The characteristic function's square modulus is $(\mathbb{E}\cos tX)^2+(\mathbb{E}\sin tX)^2=1-\operatorname{Var}\cos tX-\operatorname{Var}\sin tX$, requiring $\cos tX,\,\sin tX$ to be constant variables, which is equivalent to $\tan\tfrac{tX}{2}$ being constant.

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You wish to show that if $f:\mathbb{R} \to \mathbb{C}$ is s.t. $|f(x)| \le 1$ for each $x$ and $|\int_\mathbb{R} f(x)d\mu(x)| = 1$ for some probability measure $\mu$, then $f$ is constant $\mu$-a.e..

By multiplying $f$ by a constant, we may assume $\int_\mathbb{R} f(x)d\mu(x) = 1$. Then, $\int_\mathbb{R} Re[f(x)]d\mu(x) = 1$ and $Re[f(x)] \le 1$ for each $x$, so we must have $Re[f(x)] = 1$ for $\mu$-a.e. $x$. It follows that $f = 1$ $\mu$-a.e..

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