 Then take the Hilbert space closure with respect to this inner product, this is the Sobolev space formula_11.The continuum GFF formula_1 on formula_13 is a Gaussian random field indexed by formula_11, i.e., a collection of Gaussian random variables, one for each formula_15, denoted by formula_16, such that the covariance structure is formula_17 for all formula_18.Such a random field indeed exists, and its distribution is unique.