Verify that for the measure space $(X, \Sigma, \mu)$ = $(\Bbb N, P(\Bbb N), \text{counting measure})$ and $f$ : $\Bbb N \to \Bbb R$, one has $\int {\vert {f} \vert ^p} d\mu = \sum_{k=1}^\infty {\vert {a_k} \vert}^p$ where $a_k=f(k)$.
I know I need to define simple functions $\phi_n$ where $\phi_n= f1_{[1,n]} = f(1)1_{\{1\}}+f(2)1_{\{2\}}+...+f(n)1_{\{n\}}$, calculate the integral of this function and apply the Monotone Convergence Theorem, but I am unsure on the specifics.
