Summary: The structure of some operator ideals \({\mathfrak A}\) defined on continuous function spaces is studied. Conditions are considered under which ``\(T\in {\mathfrak A}\)'' and ``the representing measure of \(T\) takes values in \({\mathfrak A}\)'' are equivalent for the scales of \(p\)-converging \((C_ p)\) and weakly-\(p\)-compact \((W_ p)\) operators. The scale \(C_ p\) is intermediate between the ideals \(C_ 1={\mathfrak U}\) (unconditionally summing operators), and \(C_ \infty={\mathfrak B}\) (completely continuous operators), which have been studied by several authors (Bombal, Cembranos, Rodríguez-Salinas, Saab). The dual scale \(W_ p\) is intermediate between the ideals \(\mathfrak K\) (compact operators) and \(W_ \infty =W\) (weakly compact operators), and the results presented have a close connection with those of Diestel, Núñez and Seifert.