Let $w:\mathbb{N}\to [0,\mathrm{\infty})$ continuous.

For each $f:\mathbb{N}\to \mathbb{C}$ such that $\sum _{n=1}^{\mathrm{\infty}}w(n)f(n)$ is absolutely convergent we define $\mathrm{\Lambda}f={\displaystyle \sum _{n=1}^{\mathrm{\infty}}w(n)f(n)}$

It is easy to prove that $\mathrm{\Lambda}$ is linear and satisfies that $\mathrm{\forall}f:f(\mathbb{N})\subset [0,\mathrm{\infty})\Rightarrow \mathrm{\Lambda}f\in [0,\mathrm{\infty})$

(I think this last property has a name but I don't know what it is)

By the Riesz representation theorem there is only one positive measure $\nu $ such that $\mathrm{\Lambda}f={\displaystyle {\int}_{\mathbb{N}}fd\nu}$. How can I find the measure $\nu $ that fulfills this property?

For each $f:\mathbb{N}\to \mathbb{C}$ such that $\sum _{n=1}^{\mathrm{\infty}}w(n)f(n)$ is absolutely convergent we define $\mathrm{\Lambda}f={\displaystyle \sum _{n=1}^{\mathrm{\infty}}w(n)f(n)}$

It is easy to prove that $\mathrm{\Lambda}$ is linear and satisfies that $\mathrm{\forall}f:f(\mathbb{N})\subset [0,\mathrm{\infty})\Rightarrow \mathrm{\Lambda}f\in [0,\mathrm{\infty})$

(I think this last property has a name but I don't know what it is)

By the Riesz representation theorem there is only one positive measure $\nu $ such that $\mathrm{\Lambda}f={\displaystyle {\int}_{\mathbb{N}}fd\nu}$. How can I find the measure $\nu $ that fulfills this property?