Since the probability of φ k is a product, the value of φ k at each separate value of k is independently Gaussian distributed. The variance of the Gaussian is ⁠ 1 / k d k ⁠, which is formally infinite, but that just means that the fluctuations are unbounded in infinite volume. In any finite volume, the integral is replaced by a discrete sum, and the variance of the integral is ⁠ V / k ⁠.