In Theorem 1 of [this][1] paper Segal stablish a relation betwen states and generatin functionals. 
He assert that in order to $\mu$ be a generating functional  must satisfy 

[![enter image description here][2]][2] 

Then, as an example he show the functional 

[![enter image description here][3]][3]

The question is. Why this functional satisfies the desidered condition?


  [1]: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/foundations-of-the-theory-of-dynamical-systems-of-infinitely-many-degrees-of-freedom-ii/600D279603237D9BC69E458758DF682A
  [2]: https://i.sstatic.net/iLJY9.png
  [3]: https://i.sstatic.net/RZDcZ.png