How to compute an integral with respect to $K(t, dk)$

Question

Let $X$ be a random variable with distribution $F$ and $K: \mathbb{R}^+\times\mathbb{N}\to[0,1]$ defined by $K(t, \{k\}):=F(t)^{k-1}(1-F(t))$. If $u:\mathbb{R}^+\times\mathbb{N}\to\mathbb{R}^+$, how does one compute something like \begin{align*} \int e^{-u(t,k)}\,K(t, dk)\,? \end{align*} I am just very confused on the notation here, any help would be greatly appreciated.

Answer 1

Your integral is simply $$\sum_{k=1}^\infty e^{-u(t,k)}F(t)^{k-1}(1-F(t)).$$