Let A be a uniform algebra on a compact Hausdorff
space X. The spectrum, or the maximal ideal space, MA, of A is given by
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00026766/resource/name/S0008414X00026766_eqn1.gif?pub-status=live)
We define the measure spectrum, SA, of A by
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00026766/resource/name/S0008414X00026766_eqn2.gif?pub-status=live)
SA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfying
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00026766/resource/name/S0008414X00026766_eqn3.gif?pub-status=live)
The concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.