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 formula_1 be a topological vector space, formula_2 the algebraic dual space and formula_3 the topological dual space. In topological vector spaces there exist three prominent σ-algebras:
The following relationship holds:
formula_11
where formula_12 is obvious.
Cylindrical σ-algebra.
Let formula_13 and formula_14 be two vector spaces in duality. A set of the form
formula_15
for formula_16 and formula_17 is called a cylinder set and if formula_18 is open, then it's an open cylinder set. The set of all cylinders is formula_19 and
formula_20
is called the cylindrical σ-algebra. The sets of cylinders and the set of open cylinders generate the same cylindrical σ-algebra.
For the weak topology formula_21 the cylindrical σ-algebra formula_22 is the Baire σ-algebra of formula_23. 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.
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 formula_24 be a topological vector space over formula_25 and formula_2 its algebraic dual space. Furthermore, let formula_27 be a vector space of linear functionals on formula_13, that is formula_29.
A set function
formula_30
is called a cylindrical measure if, for every finite subset formula_31 with formula_32, the restriction
formula_33
is a σ-additive function, i.e. formula_34 is a measure.
Let formula_35. A cylindrical measure formula_36 on formula_13 is said to have weak order formula_38 (or to be of weak type formula_38) if the formula_38-th weak moment exists, that is,
formula_41
for all formula_42.
Radon measure.
Every Radon measure induces a cylindrical measure but the converse is not true. Let formula_43 and formula_44 be two locally convex space, then an operator formula_45 is called a formula_46-radonifying operator, if for a cylindrical measure formula_36 of order formula_48 on formula_43 the image measure formula_50 is a Radon measure of order formula_38 on formula_44.
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 formula_53 be a balanced, convex, bounded and closed subset of a locally convex space formula_43, then formula_55 denoted the subspace of formula_43 which is generated by formula_53. A balanced, convex, bounded subset formula_53 of a locally convex Hausdorff space formula_43 is called a Hilbert set if the Banach space formula_55 has a Hilbert space structure, i.e. the norm formula_61 of formula_55 can be deduced from a scalar product and formula_55 is complete.
A theorem by Sazonov-Badrikian.
Let formula_43 be a quasi-complete locally convex Hausdorff space and formula_65 be its dual equipped with the topology of uniform convergence on compact subsets in formula_43 . Assume that every subset of formula_43 is contained in a balanced, convex, compact Hilbert set. A function of positive type formula_68 on formula_65 is the Fourier transform of a Radon measure on formula_43 if and only if the function is continuous for the Hilbert-Schmidt topology associated with the topology of formula_65.
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.
Bibliography.
A valid standard reference is still the book published by Laurent Schwartz in 1973.