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Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). It is important to notice that cylinder set measures are also supported in an extremely "small" subset $\mathscr{V}\subset\mathscr{W}$, in the sense that $\mathscr{V}$ is locally compact (therefore, essentially "finite-dimensional") with respect to the induced topology from $\mathscr{W}$.

An important special case is when $\mathscr{W}$ is a nuclear space with respect to its topology, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). It is important to notice that cylinder set measures are also supported in an extremely "small" subset $\mathscr{V}\subset\mathscr{W}$, in the sense that $\mathscr{V}$ is locally compact (therefore, essentially "finite-dimensional") with respect to the induced topology from $\mathscr{W}$.

An important special case is when $\mathscr{W}$ is the dual of a nuclear space, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). It is important to notice that cylinder set measures are also supported in an extremely "small" subset $\mathscr{V}\subset\mathscr{W}$, in the sense that $\mathscr{V}$ is locally compact (therefore, essentially "finite-dimensional") with the induced topology from $\mathscr{W}$. This expresses the fact that, in infinite dimensions, the Borel $\sigma$-algebra of $\mathscr{W}$ is much larger than its cylinder set $\sigma$-algebra.

An important special case is when $\mathscr{W}$ is a nuclear space with respect to its toplogy, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). It is important to notice that cylinder set measures are also supported in an extremely "small" subset $\mathscr{V}\subset\mathscr{W}$, in the sense that $\mathscr{V}$ is locally compact (therefore, essentially "finite-dimensional") with respect to the induced topology from $\mathscr{W}$.

An important special case is when $\mathscr{W}$ is a nuclear space with respect to its topology, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). An important special case is when $\mathscr{W}$ is a nuclear space, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures in $\mathscr{S}'$ which are supported in a subspace $\mathscr{W}$ which is endowed with a stronger topology which makes $\mathscr{W}$ a Polish space, that is, a separable complete metric space (in particular, second countable). This is the natural context for cylinder set measures, as discussed in Bourbaki's "Eléments de Mathématiques, Intégration" (Hermann, 1969), Chapter 9 (there, such measures are called "promeasures", for they can be understood as "projective systems of probability measures"). It is important to notice that cylinder set measures are also supported in an extremely "small" subset $\mathscr{V}\subset\mathscr{W}$, in the sense that $\mathscr{V}$ is locally compact (therefore, essentially "finite-dimensional") with the induced topology from $\mathscr{W}$. This expresses the fact that, in infinite dimensions, the Borel $\sigma$-algebra of $\mathscr{W}$ is much larger than its cylinder set $\sigma$-algebra.

An important special case is when $\mathscr{W}$ is a nuclear space with respect to its toplogy, which is discussed in Gel'fand-Vilenkin's "Generalized Functions", Volume 4. You can also find material on this in the book of J. Glimm and A. Jaffe, "Quantum Physics - A Functional Integral Point of View" (2nd. ed., Springer-Verlag, 1988).