In the Wightman setting a quantum field is defined as an operator valued distribution satisfying the Wightman axioms. Using the vacuum state of this theory we can define correlation/Wightman functions that are known to satisfy certain properties.

We can also work in the opposite direction. Given a set of scalar-valued distributions $\{W_n\}$ obeying certain properties there is an associated unique quantum field theory (in other words, a unique set of operator-valued distributions obeying the Wightman axioms for which the $W_n$ are its correlation functions) thanks to the Wightman reconstruction theorem.

If we are given a measure $\mu$ then we can of course define correlation functions by taking moments of $\mu$ and check if these moments satisfy the conditions needed to invoke the Wightman reconstruction theorem. However the reconstruction theorem itself makes no mention of a measure $\mu$, so $\{W_n\}$ doesn't necessarily have to be moments of some measure.

At what point does the measure $\mu$ enter? Do we always consider a measure because of motivation from physics and the path integral? That is, when rigorously constructing quantum field theories do we always study correlation functions arising from a measure because these are the only theories that are physically relevant? I am trying to distinguish the physics and the mathematics behind these constructions.