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You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know can be found in Laurent Schwartz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covariance operator $\mathscr{K}$.

In any case, have a look at the above two references.

Edit: The book Gaussian measures by Bogachev is also a very good source.

You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know can be found in Laurent Schwartz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covariance operator $\mathscr{K}$.

In any case, have a look at the above two references.

Edit: The book Gaussian measures by Bogachev is also a very good source.

2023 Edit There is a new (2023) book by Stroock Gaussian measures in Finite and Infinite Dimensions that does a particularly good job of explaining Gaussian measures and some and their uses. It is not as general as Bogachev's book (only deals with Banach spaces) but it uncovers a lot of the "mystery" of this concept.

You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know can be found in Laurent Schwarz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covariance operator $\mathscr{K}$.

In any case, have a look at the above two references.

Edit: The book Gaussian measures by Bogachev is also a very good source.

You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know can be found in Laurent Schwartz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covariance operator $\mathscr{K}$.

In any case, have a look at the above two references.

Edit: The book Gaussian measures by Bogachev is also a very good source.

You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know cand be found in Laurent Schwarz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covaraince operatro $\mathscr{K}$.

In any case, have a look at the above two references.

You ought to have a look at the $4$-th volume of Gelfand-Vilenkin on Generalized Functions where they describe this concept in great detail, albeit in an old-fashion language. The most comprehensive description I know can be found in Laurent Schwarz' book Radon measures.

Things are pretty reasonable for Gaussian measures defined on duals of nuclear spaces. The space of distributions (generalized functions) on an domain of $\mathbb{R}^n$ is such a space. The Wiener measure is defined on a space of generalized functions, but it is supported on a much "thinner" space.

Beyond duals of nuclear spaces you need to assume some things about the covariance operator $\mathscr{K}$.

In any case, have a look at the above two references.

Edit: The book Gaussian measures by Bogachev is also a very good source.