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Let $X$ be a random variable defined on a probability space $(\Omega,\mathcal{F}, P)$ and let $\varphi_X$ denote the charateristic function of $X$. Suppose there exists a $t_0 \in \mathbb{R}$ \ $\{0\}$ s.t. $\varphi_X(t_0)=1$.

Show that $X$ is a discrete random variable.


Thoughts: From the definition in my course book, $X$ is discrete if $P(X \in R_X) =1$ where $R_X = \{ x \in \mathbb{R} : P(X=x)>0 \}$.

Knowing that $\varphi_X(t_0)=1$, I have $E[e^{it_0X}]=\int_\Omega e^{it_0X}dP=1 \implies \int_\Omega e^{it_0X}-1dP=0$.

However I am not sure how to connect this, and even if I am supposed to something that tells me $X$ is discrete. Any help is appreciated!

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Note that $1 = \phi_X(t_0) = \mathbb E[\exp(it_0X)] = \mathbb E[\cos(t_0X)]$ (since imaginary party must be $0$).

That means: $0 = \int_{\mathbb R} 1-\cos(t_0x) d\mu_X(x) $

But since $1-\cos(t_0x) \ge 0$, we must have $1 = \cos(t_0x)$ but $\mu_X $- almost surely

Which means that $\mu_X$ must be concentrated on the set $\{2k\pi\frac{1}{t_0} : k \in \mathbb Z\}$

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