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For the given exchangeable law $\mu$, putwhich is well-defined by the stated assumption. Integrate the product self-duality of symmetric exclusion against $\mu$, and use boundedness to interchange expectation and integration:The finite-particle process conserves its cardinality. Hence the right side is ${\mathbb{E}}^A{m}_{|A_t|}=m_{|A|}$. All occupation-product moments are therefore unchanged by evolution.

These moments determine the law, not merely its one-site marginals. For disjoint finite sets $B,C$, the inclusion-exclusion principle givesEvery term is unchanged, so every finite cylinder probability is unchanged. Such cylinder events generate the product sigma-algebra on $\{0,1{\}}^{\mathbb{Z}}$, and probability laws agreeing on them agree everywhere. ConsequentlyThus every exchangeable probability law is invariant for the symmetric exclusion process. This proof of exchangeable invariant laws of symmetric exclusion needs only the given cardinality dependence and duality; no representation theorem for exchangeable measures is required.