Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding.
In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous
construction I mean a construction of the quadruple $(\mathcal{H},U,\Omega,\phi)$ where $\mathcal{H}$ is
the Hilbert space of states, $\Omega$ the vacuum vector, $U$ a unitary representation and $\phi$ an
operator valued distribution. These data have to satisfy certain axioms which are called the *Wightman* axioms.
However this program of explicitly constructing QFT turned out to be too difficult and the other strategies
emerged: one of them is concerned with the construction of the functional integral, i.e. the problem
boils down to the construction of certain measure on the space of distributions. However any interacting
QFT is governed by a Lagrangian: I don't see where exactly this Lagrangian enters in the above reasoning.
So to be slightly more precise:

Question 1. Suppose that we want to construct QFT following the „original approach” (i.e. constructing it directly). Which one of the Wightman axioms tells us which QFT we have really constructed (i.e. what is the form of the interaction part of a Lagrangian)?

And concerning the functional integral approach:

Question 2. Is there a precise form/conditions for a desired measure (which can be read from the Lagrangian) or is it given only at the heuristic level via some density function with respect to a Gaussian measure (but in the end this measure may turn out to be singular to a Gaussian measure)? If it is given only on a heuristic level how it is possible to know whether we have succeed in our construction?

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