Minlos–Sazonov theorem

The Minlos–Sasonov theorem is a result from measure theory in topological vector spaces. It provides a sufficient condition for a cylindrical measure to be σ-additive on a locally convex space. This is the case when its Fourier transform is continuous at zero in the Sazonov topology and such a topology is called sufficient. The theorem is named after the two Russian mathematicians Robert Adol'fovich Minlos and Vyacheslav Vasilievich Sazonov.

The theorem generalizes two classical theorem: the Minlos theorem (1963) and the Sazonov theorem (1958). It was then later generalized in the 1970s by the mathematicians Albert Badrikian and Laurent Schwartz to locally convex spaces. Therefore, the theorem is sometimes also called Minlos-Sasonov-Badrikian theorem.^[1][2]

Let ${\displaystyle (X,\tau )}$ be a locally convex space, ${\displaystyle X^{*}}$ and ${\displaystyle X'}$ are the corresponding algebraic and topological dual spaces, and ${\displaystyle \langle ,\rangle :X\times X'\to \mathbb {R} }$ is the dual paar. A topology ${\displaystyle \tau ^{K}}$ on ${\displaystyle X}$ is called compatible with the dual paar ${\displaystyle \langle ,\rangle }$ if the corresponding topological dual space is ${\displaystyle X'}$. A seminorm ${\displaystyle p}$ on ${\displaystyle X}$ is called Hilbertian or a Hilbert seminorm if there exists a positive definite bilinear form ${\displaystyle b\colon X\times X\to \mathbb {R} }$ such that ${\displaystyle p(x)={\sqrt {b(x,x)}}}$ for all ${\displaystyle x\in X}$.

Let ${\displaystyle {\mathfrak {A}}:={\mathfrak {A}}(X,X'):=\bigotimes \limits _{n=1}^{\infty }{\mathfrak {A}}_{f_{1},\dots ,f_{n}}}$ denote the cylindrical algebra.^[3]

Let ${\displaystyle p}$ be a seminorm on ${\displaystyle X}$ and ${\displaystyle X_{p}}$ be the factor space ${\displaystyle X_{p}:=X/p^{-1}(0)}$ with canonical mapping ${\displaystyle Q_{p}:X\to X_{p}}$ defined as ${\displaystyle Q_{p}:x\mapsto [x]}$. Let ${\displaystyle {\overline {p}}}$ be the norm

${\displaystyle {\overline {p}}(y)=p\left(Q_{p}^{-1}(y)\right)}$

on ${\displaystyle X_{p}}$, denote the corresponding Banach space as ${\displaystyle {\overline {X}}_{p}}$ and let ${\displaystyle i_{p}:X_{p}\hookrightarrow {\overline {X}}_{p}}$ be the natural embedding, then define the continuous map

${\displaystyle {\overline {Q}}_{p}(x):=i_{p}\left(Q_{p}(x)\right)}$

which is a map ${\displaystyle {\overline {Q}}_{p}:X\to {\overline {X}}_{p}}$. Let ${\displaystyle q}$ be a seminorm such that for all ${\displaystyle x\in X}$

${\displaystyle p(x)\leq Cq(x),}$

then define a continuous linear operator ${\displaystyle T_{q,p}:{\overline {X}}_{q}\to {\overline {X}}_{p}}$ as follows:

• If ${\displaystyle z\in i_{q}(X_{q})\subseteq {\overline {X}}_{q}}$ then ${\displaystyle T_{q,p}(z):={\overline {Q}}_{p}\left({\overline {Q}}_{q}^{-1}(z)\right)}$, which is well-defined.
• If ${\displaystyle z\not \in i_{q}(X_{q})}$ and ${\displaystyle z\in {\overline {X}}_{q}}$, then there exists a sequence ${\displaystyle (z_{n})_{n}\in i_{q}(X_{q})}$ which converges against ${\displaystyle z}$ and the sequence ${\displaystyle \left(T_{q,p}(z_{n})\right)_{n}}$ converges in ${\displaystyle {\overline {X}}_{p}}$ therefore ${\displaystyle T_{q,p}(z):=\lim \limits _{n\to \infty }\left(T_{q,p}(z_{n})\right)_{n}.}$^[4]

If ${\displaystyle p}$ it Hilbertian then ${\displaystyle {\overline {X}}_{p}}$ is a Hilbert space.

Let ${\displaystyle {\mathcal {P}}}$ be a family of continuous Hilbert seminorms defined as follows: ${\displaystyle p\in {\mathcal {P}}}$ if and only if there exists a Hilbert seminorm ${\displaystyle q}$ such that for all ${\displaystyle x\in X}$

${\displaystyle p(x)\leq Cq(x)}$

for some constant ${\displaystyle C\in \mathbb {R} }$ and if ${\displaystyle T_{q,p}}$ is a Hilbert-Schmidt operator. Then the topology ${\displaystyle \tau ^{S}:=\tau ^{S}(X,\tau )}$ induced by the family ${\displaystyle {\mathcal {P}}}$ is called the Sazonov topology or S-Topologie.^[4] Clearly it depends on the underlying topology ${\displaystyle \tau }$ and if ${\displaystyle (X,\tau )}$ is a nuclear then ${\displaystyle \tau ^{S}=\tau }$.

Let ${\displaystyle \mu }$ be a cylindrical measure on ${\displaystyle {\mathfrak {A}}}$ and ${\displaystyle \tau }$ a locally convex topology that is compatible with the dual paar and let ${\displaystyle \tau ^{S}:=\tau ^{S}(X,\tau )}$ be the Sazonov topology. Then ${\displaystyle \mu }$ is σ-additive on ${\displaystyle {\mathfrak {A}}}$ if the Fourier transform ${\displaystyle {\hat {\mu }}(f):X'\to \mathbb {C} }$ is continuous in zero in ${\displaystyle \tau ^{S}}$.^[4]

• Schwartz, Laurent (1973). Radon measures on arbitrary topological spaces and cylindrical measures. Tata Institute of Fundamental Research Studies in Mathematics.
• Bogachev, Vladimir I.; Smolyanov, Oleg G. (2017). Topological Vector Spaces and Their Applications. Springer Cham.