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We consider the set $S\left(\mu \right)$ of complex-valued homomorphisms of a uniform algebra $A$ which are weak-star continuous with respect to a fixed measure $\mu$. The $\mu$-parts of $S\left(\mu \right)$ are defined, and a decomposition theorem for measures in $A^{\perp }\cap L^1\left(\mu \right)$ is obtained, in which constituent summands are mutually absolutely continuous with respect to representing measures. The set $S\left(\mu \right)$ is studied for $T$-invariant algebras on compact subsets of the complex plane and also for the infinite polydisc algebra.