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I would like to know if there exists a measure $\rho$ on the positive real line such that its moments $\int_0^{\infty} x^j d\rho(x)$ are equal to a constant (for example equal to one) for all $j=0,\dots,n,\dots$ (or for $j\leq n$ for any $n$). In other words, if there exists a density function (equal to zero on the negative real part) such that its characteristic function is $e^{it}$.

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    Got something from an answer below?2012-07-25

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(With no characteristic function.)

Note that $\int(x-1)^2\,\mathrm d\rho(x)=\int x^2\,\mathrm d\rho(x)-2\int x\,\mathrm d\rho(x)+\int 1\,\mathrm d\rho(x). $ Hence, if $ \int x^2\,\mathrm d\rho(x)=\int x\,\mathrm d\rho(x)=\int 1\,\mathrm d\rho(x), $ then the set $\mathbb R\setminus\{1\}$ has measure zero with respect to $\rho$. (No hypothesis on some other moments is necessary.)

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The (unique) random variable whose characteristic function is $t\mapsto e^{it}$ is associated to the measure $\delta_1$, that is $\delta_1(A)=\begin{cases} 1&\mbox{ if }1\in A,\\ 0&\mbox{ otherwise}. \end{cases}$ This is not a continuous random variable.