Measure theory in topological vector spaces


In mathematics, measure theory in topological vector spaces refers to the extension of measure theory to topological vector spaces. Such spaces are often infinite-dimensional, but many results of classical measure theory are formulated for finite-dimensional spaces and cannot be directly transferred. This is already evident in the case of the Lebesgue measure, which does not exist in general infinite-dimensional spaces.
The article considers only topological vector spaces, which also possess the Hausdorff property. Vector spaces without topology are mathematically not that interesting because concepts such as convergence and continuity are not defined there.

σ-Algebras

Let be a topological vector space, the algebraic dual space and the topological dual space. In topological vector spaces there exist three prominent σ-algebras:
The following relationship holds:
where is obvious.

Cylindrical σ-algebra

Let and be two vector spaces in duality. A set of the form
for and is called a cylinder set and if is open, then it's an open cylinder set. The set of all cylinders is and the set of all open cylinders is. If one takes the product, this only produces an algebra. The σ-algebra
is called the cylindrical σ-algebra. The sets of cylinders and the set of open cylinders generate the same cylindrical σ-algebra, i.e..
For the weak topology the cylindrical σ-algebra is the Baire σ-algebra of. One uses the cylindrical σ-algebra because the Borel σ-algebra can lead to measurability problems in infinite-dimensional space. In connection with integrals of continuous functions it is difficult or even impossible to extend them to arbitrary borel sets. For non-separable spaces it can happen that the vector addition is no longer measurable to the product algebra of borel σ-algebras because in general, however for the cylindrical σ-algebra one has.

Equality of the σ-algebras

  • Let be a topological vector space and let be the weak topology, then is exactly the Baire σ-algebra of.
  • Let be a separable, metrizable locally convex space and be the weak topology. Then, and are equivalent under and.

Measures

One way to construct a measure on an infinite-dimensional space is to first define the measure on finite-dimensional spaces and then extend it to infinite-dimensional spaces as a projective system. This leads to the notion of cylindrical measure, which according to Israel Moiseevich Gelfand and Naum Yakovlevich Vilenkin, originates from Andrei Nikolayevich Kolmogorov.

Cylindrical Measures

Let be a topological vector space over and its algebraic dual space. Furthermore, let be a vector space of linear functionals on, that is.
A set function
is called a cylindrical measure if, for every finite subset with, the restriction
is a σ-additive function, i.e. is a measure.
Let. A cylindrical measure on is said to have weak order if the -th weak moment exists, that is,
for all.

Radon measure

Every Radon measure induces a cylindrical measure but the converse is not true. Let and be two locally convex space, then an operator is called a -radonifying operator, if for a cylindrical measure of order on the image measure is a Radon measure of order on.

Some results

There are many results on when a cylindrical measure can be extended to a Radon measure, such as Minlos theorem and Sazonov theorem.
Let be a balanced, convex, bounded and closed subset of a convex space, then denoted the subspace of which is generated by. A balanced, convex, bounded subset of a locally convex Hausdorff space is called a Hilbert set if the Banach space has a Hilbert space structure, i.e. the norm of can be deduced from a scalar product and is complete.

A theorem by Sazonov-Badrikian

Let be a quasi-complete locally convex Hausdorff space and be its dual equipped with the topology of uniform convergence on compact subsets in . Assume that every subset of is contained in a balanced, convex, compact Hilbert set. A function of positive type on is the Fourier transform of a Radon measure on if and only if the function is continuous for the Hilbert-Schmidt topology associated with the topology of.

Minlos–Sasonov theorem

A slight variant of the theorem is the Minlos–Sazonov theorem which states that a cylindrical measure is σ-additive and Radon if it's Fourier transform is continuous in zero in the Sazonov topology.