Definition 16.1.1 (Space of Simple Functions). Let $(X, \cm)$ be a measurable space, then $\Sigma(X, \cm) = \Sigma(X, \cm; \complex)$ is the space of $\complex$-valued simple functions on $(X, \cm)$, and $\Sigma^{+}(X, \cm)$ is the space of non-negative simple functions.