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Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?

In this article Sourav Chatterjee poses the question, how do we define the measure: $$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$ The $Z$ here is an infinite normalizing ...
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