# Tagged Questions

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### $L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
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### Writing $f\in L^2([-\pi,\pi])$ as a power series.

Consider the space $L^2([-\pi,\pi])$. I want to show that every function $f\in L^2([-\pi,\pi])$ can be written as a power series. I remember a result that polynomials are dense in $L^2([-\pi,\pi])$. ...
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### The radius of convergence of the power series is a constant with probability one.

I came across this question. If $\{X_n , n\geq 1\}$ are independent random variables, show that the radius of convergence of the power series $\sum_{n=1}^\infty X_n z^n$ is a constant with probability ...
### Show that Laplace transform of measure belongs to $C^{\infty}(0,\mathbb{R}^n_{+})$
Let $\mu$ be an exponentially decreasing Borel measure on $\mathbb{R}^n_{+}$, i.e. there exists $r>0$ such that $$\int\limits_{\mathbb{R}^n_{+}} e^{r|x|} \, \mu(dx) < \infty.$$ I want to ...
Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, ...