We investigate operator ideal properties of convolution operators $C_λ $ (via measures $λ$) acting in ${L^∞ (G)}$, with $G$ a compact abelian group. Of interest is when $C_λ$ is compact, as this corresponds to $λ$ having an integrable density relative to Haar measure $μ$, i.e., $λ ≪ μ $. Precisely then is there an \textit{optimal} Banach function space $L^1 (m_λ)$ available which contains ${L^∞ (G)}$ properly, densely and continuously and such that $C_λ$ has a continuous, ${L^∞ (G)}$-valued, linear extension $I_{m_λ}$ to $L^1 (m_λ)$. A detailed study is made of $L^1 (m_λ)$ and $I_{m_λ}$. Amongst other things, it is shown that $C_λ$ is compact iff the finitely additive, ${L^∞ (G)}$-valued set function $m_λ (A) := C_λ ({χ_{_{_{\scriptstyle{A}}}}})$ is norm $σ$-additive iff $λ 𝟄 L^1 (G)$, whereas the corresponding optimal extension $I_{m_λ}$ is compact iff $λ 𝟄 C (G)$ iff $m_λ$ has finite variation. We also characterize when $m_λ$ admits a Bochner (resp.\ Pettis) $μ$-integrable, $L^{∞} (G)$-valued density.