Nonnegative scalar multiples and countable weighted sums of measures

Let $\mu$ be a measure and let $c\in [0,+\mathrm{\infty }]$. Define the scalar set function $c\mu$ by three disjoint branches:

$(c\mu )(E):=\{\begin{array}{ll}0,& c=0,\\ c\mu (E),& 0<c<+\mathrm{\infty },\\ 0,& c=+\mathrm{\infty }\text{ and }\mu (E)=0,\\ +\mathrm{\infty },& c=+\mathrm{\infty }\text{ and }\mu (E)>0.\end{array}$

Thus neither the zero branch nor the infinite branch forms the undefined extended-real product $0\cdot (+\mathrm{\infty })$.

For measures $({\mu }_k{)}_{k\in \mathbb{N}}$ on the same measurable space and weights $c_k\in [0,+\mathrm{\infty }]$, their countable weighted sum is the pointwise set function

$\left({\sum }_{k=0}^{\mathrm{\infty }}{c}_k{\mu }_k\right)(E):={\sum }_{k=0}^{\mathrm{\infty }}(c_k{\mu }_k)(E),$

using Series in the nonnegative extended real line. Finite weighted sums are defined by the corresponding finite partial sum, with the empty weighted sum the zero set function. The fact that all these set functions are measures is Nonnegative scalar multiples and countable weighted sums of measures are measures ↗.