Let $X$ be a complex Banach space and let $M$ be a closed subspace of $L^{\infty}(J,X)$, where $J\in \{\mathbf{R},\mathbf{R}^+\}$. We answer the following question: Under what conditions $\phi_s-\phi\in M \ \forall s\in J$ implies that $\phi\in M$. Some conditions will be imposed on $M$ to obtain the main result concerning the indefinite integral. These conditions guarantee the following implication: $F\in E(J,X)\Longrightarrow F\in M$, where $F$ is the integral $\int_0^t f(s)ds$ of $f\in M\cap C_{ub}(J,X)$. Also, we generalize Loomis' Theorem for almost periodic functions [19, Theorem 5], to a more general class of functions $M\subseteq L^{\infty}(\mathbf{R},X)$ containing $AP(\mathbf{R},X)$. The main result of Part IV is: If $\phi$ is uniformly continuous, bounded, such that the $M$-spectrum $\sigma_M(\phi)$ of $\phi$ is at most countable and, for every $\lambda\in\sigma_M(\phi)$, the function $e^{-i\lambda t}\phi(t)$ is ergodic, then $\phi\in M$.