Here is my solution for this: For any $k\in \mathbb N$, $\{1\}, \{2\},..., \{k\}$, $Z^+- \{1,2,...,k\}$ is a partition $P$ of $Z^+$. So we have $L(b, P)=\sum_{i=1}^k b_i$. It follows that $\int f dc\ge \sum_{i=1}^k b_i$ for all $k$. Letting $k\to \infty$, we get $\int f dc\ge \sum_i b_i$.

For the reverse inequality, let $P=\{A_1,...,A_k\}$ be any partition of $Z^+$. We have $L(b, P)= \sum_{i=1}^k (\inf_{A_i} b) c(A_i)\le \sum_{i=1}^\infty b_i$, where the last inequality is by non negativity of $b_i$'s. It follows that $\int f\, dc\le \sum_{i=1}^\infty b_i.$