On a Hilbert space we define a probabilistic measure with expected value and covariance operator. Then Gaussian measure is defined on real line and later extended to Hilbert space by use of Fourier transform of measure. Notion of product measure helps us compute some integrals with respect to Gaussian measure. Properties of Gaussian measures are passed to Gaussian random variables with values in Hilbert space and some examples of these variables are presented. We investigate conditions for their indendence and put them in perspective of spaces of equivalence classes. At last we define a white noise function, which takes an element of Hilbert space and returns a random variable defined on a same Hilbert space. We use this result to construct a Brownian motion.
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