how to show $\int|f|^{p} \mathrm{d} \mu=\sum_{n=1}^{\infty} \frac{\sqrt{n}^{p}}{n^{2}}$?

Question

Let $(X,\mathbb{X}, \lambda)$ be a measure space.

Question

Let $X=\mathbb{N}$ and let $\mathbb{X}$ be the collection of all subsets of $\mathbb{N}$. Let $\lambda$ be defined by $$ \lambda(E)=\sum_{n \in E}\left(1 / n^{2}\right), E \in \mathbb{X} $$ (a) Show that $f: X \rightarrow \mathbb{R}, f(n)=\sqrt{n}$ satisfies $f \in L_{p}$ if and only if $1 \leq p<2 .$

Solution

We have that $$ \int|f|^{p} \mathrm{d} \mu=\sum_{n=1}^{\infty} \frac{\sqrt{n}^{p}}{n^{2}}=\sum_{n=1}^{\infty} n^{p / 2-2} $$ Thus $f \in L_{p}$ if and only if $(p / 2-2)<-1$ which happens if and only if $1 \leq p<2$

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I don't get how the first equality if obtained. how do we find? $$\int|f|^{p} \mathrm{d} \mu=\sum_{n=1}^{\infty} \frac{\sqrt{n}^{p}}{n^{2}}$$

I tried with definition of Lebesgue integral by trying to find an approximation by a simple function $\phi$ of the function $f$, but I can't find $\phi$ with this answer.

Answer 1

The measure $\lambda$ is a point mass distribution on the natural numbers so the integral $\int_{\Bbb{N}}|f(x)|^pd\lambda(x)=\sum_{n=1}^{\infty}|f(n)|^p\frac{1}{n^2}$

Take a look at these notes to see the definition of the integral with respect to a point mass distribution.

http://fourier.math.uoc.gr/~papadim/measure_theory.pdf