We investigate operator ideal properties of convolution operators Cλ (via measures λ) acting in L∞(G), with G a compact abelian group. Of interest is when Cλ 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 L1(mλ) available which contains L∞(G) properly, densely and continuously and such that Cλ has a continuous, L∞(G)-valued, linear extension Imλ to L1(mλ). A detailed study is made of L1(mλ) and Imλ. Amongst other things, it is shown that Cλ is compact iff the finitely additive, L∞(G)-valued set function mλ(A):=Cλ(χA) is norm σ-additive iff λ∈L1(G), whereas the corresponding optimal extension Imλ is compact iff λ∈C(G) iff mλ has finite variation. We also characterize when mλ admits a Bochner (resp.\ Pettis) μ-integrable, L∞(G)-valued density