Let $\mu$ be the counting measure on $\Bbb N$ whose sigma-algebra is its power set. Show that $L^1(\Bbb N)$ is the space of all real sequences ($a_n$) whose infinite sums are absolutely convergent.
I know that the problem is the same as asking if $L^1(\Bbb N)=\{(a_n):\sum_{n=0}^\infty |a_n|\lt \infty \rbrace$. But I'm not sure how to move on, could someone provide a proof please? Thanks.
By definition, if $f \in L^1(\mathbb{N})$, then $$ \int |f|\,d\mu < \infty $$ We also have $$ \int |f|\,d\mu = \sup\left\{\int \phi\,d\mu \mid \phi \leq |f|, \phi \in \mathcal S \right\} $$ where $\mathcal S$ is the set of simple functions. In the context of counting measure, any $\phi \in \mathcal S$ corresponds to a sequence whose all but finitely many elements are zero. Observe that if $\phi = \sum_{j=1}^m a_j \chi_{\{j\}}$, then $$ \sum_{j=1}^m |f(j)| = \int \phi\,d\mu $$ so each $\phi \in \mathcal S$ corresponds to a partial sum of $|f(j)|$ with some element set to $0$. Since $\sum_{j=1}^m |f(j)|$ increases monotonically as $m$ increases, $$\sup\left\{\int \phi\,d\mu \mid \phi \leq |f|, \phi \in \mathcal S \right\} = \lim_{m \to \infty} \sum_{j=1}^m |f(j)| < \infty $$ as desired.