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Take a look at Appendix A of the 2nd edition of Glimm & Jaffe's book. They give a rigorous construction of the measure you're after aka, the Ornstein-Uhlenbeck measure, which is the cylinder measure you get by taking the continuum limit of the Euclidean signature harmonic oscillator). The key point is that the cylinder measure defined using the momentum basis (or the lattice approximation) 'vanishes at infinity', hence is an actual measure.

They also prove that this measure is supported on (distributions almost everywhere equal to) continuous functions.