Definition 1

Let $\mu$ be a positive measure on T. The pair $(\pi, g)$ is said to be $\mu$-adapted if the following conditions are satisfied: 1° The functions $\pi$ and $g$ are $\mu$-measurable. 2° For every function $f \in \mathscr{K}(X)$, the mapping $t \mapsto f(\pi(t))g(t)$ is essentially $\mu$-integrable.