Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 396–401. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 397–402. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Temirgaliev. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

On the Construction of Probability Measures on Functional Classes

N. T. Temirgaliev

Received February, 1997

1. The problem of introduction of measures on functional classes, besides being of interests in itself, has numerous applications, in particular, it is used for comparing, in the same terms, the determinate methods of numerical integration and Monte–Karlo methods (see, e.g., [1]–[7]).

Obviously, the most well-known of the measures on classes of functions is Wiener’s measure (see, e.g., [8]) defined on the class of all functions f(x), continuous on the interval [0, 1], such that f(0) = 0.

Wiener’s measure is concentrated on the set of nowhere differentiable continuous functions, in the scale of Hölder classes which have smoothness equal to 1/2 (to be more precise, the measurable set Lip α has measure 1 or 0 according as 0 < α < 1/2 or 1/2 < α < 1).

However, in different problems of the theory of functions (quadrature formulas, restoration operators, widths, etc.) classes are studied which reflect certain difference-differential properties of functions.

In this article Kolmogorov’s general theorem on the extension of measure [9] (formulated as two theorems for convenience) is defined concretely to serve as the basis for the presentation of efficient methods of constructing probability measures on a number of functional classes which are important in the theory of functions and in other fields of mathematics.

Let us now pass to constructions.

2. Suppose that we are given a number s (s = 1, 2, …) and that {Γ_{k}}_{k=0}^{∞} is a sequence of pairwise nonintersecting finite sets Γ_{k} ⊂ ℤ^{s} whose union is all ℤ^{s}. We denote by d_{k} the number of points in Γ_{k}. Let ν_{−1} = 0 and ν_{k} = d_{0} + d_{1} + ⋯ + d_{k}, and let j(m) be fixed ordering of Γ_{k}. Then every collection (the s-dimensional sequence)

Y = {y_{m}}_{m∈ℤ^{s}}  (1)

of complex numbers, with due account of the relation y_{m} = a_{j(m)} + ib_{j(m)}, can be considered to be represented as the sequence Y = (a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}, …). Finally, suppose that for every k on the 2d_{k}-dimensional Euclidean space ℝ^{2d_{k}} a nonnegative continuous function Ψ_{k} is defined such that Ψ_{k}(0) = 0. Let us define the classes H(Γ_{k}, Ψ_{k}) as the set of all collections (1) such that the inequality

Ψ_{k}(a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) ≤ 1

is satisfied for every k (k = 0, 1, 2, …).

Let

D_{k} = {(a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ ℝ^{2d_{k}} : Ψ_{k}(a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) ≤ 1}.

ON THE CONSTRUCTION OF PROBABILITY MEASURES ON FUNCTIONAL CLASSES 397

We denote by L(D_{k}) the σ-algebra of all Lebesgue measurable subsets D_{k} ⊂ ℝ^{2d_{k}}.

We define the measurable space (H, F(H)) as the least σ-algebra containing the algebra L(H) of all cylindrical sets (k = 0, 1, 2, …, E_{k} ∈ L(D_{k})):

T_{k}(E_{k}) ≡ {(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}, …) ∈ H(Γ_{k}, Ψ_{k}) : (a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ E_{k}}.

The following theorem is valid.

Theorem 1. Every one of the various probability measures μ on the measurable space (H, F(H)) is uniquely defined by the specification of the sequence of probability measures μ_{k} on (D_{k}, L(D_{k})) such that the relation

μ(T_{k}(E_{k})) = μ_{k}(E_{k})

holds for all k (k = 0, 1, 2, …) and E_{k} (E_{k} ∈ L(D_{k})).

As was pointed out above, this statement is a concretization of the theorem on the infinite product of probability measures (D_{k}, L(D_{k}), μ_{k}) (see, e.g., [10, pp. 152–156]).

Now we denote by B(Γ_{k}, Ψ_{k}) the set of all collections (1) such that

∑_{n=0}^{∞} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}}) ≤ 1.  (2)

Let

D_{k} = {(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) : ∑_{n=0}^{k} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}}) ≤ 1}

and L(D_{k}) be the σ-algebra of all subsets D_{k} measurable in the sense of 2ν_{k}-dimensional Lebesgue measure.

We define the measurable space (B, F(B)) as the least σ-algebra containing the algebra L(B) of cylindrical sets (k = 0, 1, 2, …, E_{k} ∈ L(D_{k}))

S(E_{k}) = {(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}, …) ∈ B(Γ_{k}, Ψ_{k}) : (a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ E_{k}}.

We have the following theorem.

Theorem 2. Every one of the probability measures μ on the measurable space (B, F(B)) is uniquely defined by the specification of the sequence of probability measures μ_{k} on (D_{k}, L(D_{k})) such that for all k (k = 0, 1, 2, …) and all E_{k} ∈ L(D_{k}), besides the relation

μ(S(E_{k})) = μ(E_{k}),

the following compatibility condition is fulfilled (l = 1, 2, …)

μ_{k}(E_{k}) = μ_{k+l}{(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) ∈ ℝ^{2ν_{k+l}} :
(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ E_{k}, ∑_{n=0}^{k+l} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}}) ≤ 1}. (3)

Since for every E_{k} ∈ L(D_{k}) the set E_{k} × ℝ^{2(ν_{k+l}−ν_{k})} is measurable in the sense of 2ν_{k+l}- dimensional Lebesgue measure as the product of measurable sets and the function

ξ(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) ≡ ∑_{n=0}^{k+l} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}})

is continuous on it, the set

{(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) ∈ ℝ^{2ν_{k+l}} : (a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ E_{k},

∑_{n=0}^{k+l} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}}) ≤ 1} =

= {(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) ∈ E_{k} × ℝ^{2(ν_{k+l}−ν_{k})} : ξ(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) ≤ 1}

is measurable in the sense of 2ν_{k+l}-dimensional Lebesgue measure.

Thus the formulation of the compatibility condition (3) is correct. In all other respects, the proof of Theorem 2 repeats the proof of Kolmogorov’s theorem on the extension of the measure into (ℝ^{∞}, B(ℝ^{∞})) from [11, pp. 178–181]) if the compatibility condition in it, where in (x_{1}, …, x_{n}, …) ∈ ℝ^{∞} the coordinates x_{i} and x_{j} for i ≠ j are independent of each other, is replaced by condition (3), where this independence no longer exists, which fact is reflected in condition (2).

In the case of measures μ_{k} which are absolutely continuous respective to the Lebesgue measure, conditions (3) become more visible which makes it possible to construct the corresponding measures. Namely, by virtue of the Radon–Nikodým theorem, for every k there exist nonnegative functions φ(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}), defined on D_{k}, measurable in the sense of the 2ν_{k}-dimensionsl Lebesgue measure, such that

μ_{k}(E_{k}) = ∫_{E_{k}} φ_{k}.  (4)

It is easy to see that the compatibility conditions (3) become the conditions

φ_{k}(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) = ∫_{A_{k}(a_{1},b_{1},...,a_{ν_{k}} ,b_{ν_{k}})}φ_{k+1}(a_{1}, b_{1}, …, a_{ν_{k+l}}, b_{ν_{k+l}}) da_{ν_{k}+1} db_{ν_{k}+1} ⋯ da_{ν_{k+1}} db_{ν_{k+1}},

(5)

where

A_{k}(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) = {a_{ν_{k}+1}, b_{ν_{k}+1}, …, a_{ν_{k+1}}, b_{ν_{k+1}}) :

Ψ_{k+1}(a_{ν_{k}+1}, b_{ν_{k}+1}, …, a_{ν_{k+1}}, b_{ν_{k+1}}) ≤ 1 − ∑_{n=0}^{k} Ψ_{n}(a_{ν_{n−1}+1}, b_{ν_{n−1}+1}, …, a_{ν_{n}}, b_{ν_{n}})}.

In turn, the functions φ_{k}, which satisfy conditions (4) and (5), can be derived by means of functions {t_{k}} such that

∫_{A_{k}(a_{1},b_{1},...,a_{ν_{k}} ,b_{ν_{k}})} t_{k+1}(a_{1}, b_{1}, …, a_{ν_{k+1}}, b_{ν_{k+1}}) da_{ν_{k}+1} db_{ν_{k}+1} ⋯ da_{ν_{k+1}} db_{ν_{k+1}} = 1  (6)

for almost all (a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) ∈ D_{k}, and then

φ_{k}(a_{1}, b_{1}, …, a_{ν_{k}}, b_{ν_{k}}) = ∏_{n=0}^{k} t_{n}(a_{1}, b_{1}, …, a_{ν_{n}}, b_{ν_{n}}),  (7)

and the fulfillment of conditions (4) and (5) is equivalent to the fulfillment of conditions (6) and (7).

Note that the case, where t_{k} does not depend on (a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) was studied by Banach [12] (see also [4]).

3. The Nikolskii–Besov classes (see [13, pp. 75–76]). Let s = 1, 2, … and k = 0, 1, 2, …,

Q_{k} = {m = (m_{1}, …, m_{s}) ∈ ℤ^{s} : −2^{k} ≤ m_{j} ≤ 2^{k}, j = 1, …, s},  Q_{−1} = ∅,  Γ_{k} = Q_{k} \ Q_{k−1}.

Here and in the sequel we denote by f̂(m) the Fourier trigonometric coefficients

f̂(m) = ∫_{[0,1]^{s}} f(x)e^{−2πi(m,x)} dx

for the functions f(x) ∈ L^{p}(0, 1)^{s}.

For the measurable function g(x) we set

∥g∥_{p}^{p} = ∫_{[0,1]^{s}} |g(x)|^{p} dx,

and for the collection of numbers {a(k)}_{k∈A}, where A is a given countable set, we denote by ∥a(k)∥_{l^{θ}(A)} sup_{k∈A} |a(k)| for θ = ∞ and (∑_{k∈A} |a(k)|^{θ})^{1/θ} for 1 ≤ θ < ∞.

The Nikolskii–Besov class B_{p,θ}^{r}(0, 1)^{s} (s = 1, 2, …, r > 0, 1 < p < ∞, 1 ≤ θ ≤ ∞) consists of all functions f(x) ∈ L^{p}(0, 1)^{s}, which are 1-periodic with respect to every variable and are such that

∥2^{kr}∥ ∑_{m∈Γ_{k}} f̂(m)e^{2πi(m,x)}∥_{p}∥_{l^{θ}(k≥0,k∈ℤ)} ≤ 1.

We set

\begin{matrix}f̂(m) = y_{m} = a_{j(m)} + ib_{j(m)}, \\ g_{k}(a_{ν_{k−1}+1}, b_{ν_{k−1}+1}, …, a_{ν_{k}}, b_{ν_{k}}) = 2^{kr}∥ ∑_{m∈Γ_{k}}(a_{j(m)} + ib_{j(m)})e^{2πi(m,x)}∥_{p}. \end{matrix}  (8)

Let Ψ_{k} = g_{k} or Ψ_{k} = g_{k}^{θ} irrespective of whether θ = ∞ or 1 ≤ θ < ∞. Then the constructions described in Theorems 1 and 2 allow us to obtain various probability measures on the classes B_{p,∞}^{r} ≡ H_{p}^{r} and B_{p,θ}^{r} respectively.

4. Nikolskii–Besov classes with a dominant mixed derivative (see, e.g., [14]). Let r = (r_{1}, …, r_{s}), r_{j} > 0, 1 < p < ∞, 1 ≤ θ ≤ ∞. The class SB_{p,θ}^{r_{1},...,r_{s}} consists of all functions f(x) ∈ L^{p}(0, 1)^{s}, which are 1-periodic with respect to every variable and are such that

∥2^{τ_{1}r_{1}+···+τ_{s}r_{s}} ∥ ∑_{2^{τ_{j} −1}−1≤|m_{j}|<2^{τ_{j}} −1} f̂(m)e^{2πi(m,x)}∥_{p}∥_{l^{θ}(τ_{j}=1,2,..., j=1,...,s)} ≤ 1.

The introduction of measures on these classes fall into the pattern from Theorems 1 and 2 for the following choice of Γ_{k} and Ψ_{k}:

Γ_{τ_{1},...,τ_{s}} = {(m_{1}, …, m_{s}) ∈ ℤ^{s} : 2^{τ_{j}−1} − 1 ≤ |m_{j}| ≤ 2^{τ_{j}} − 1},

Ψ_{τ_{1},...,τ_{s}} = g for θ = ∞ and Ψ_{τ_{1},...,τ_{s}} = g^{θ} for 1 ≤ θ < ∞, where

g = 2^{τ_{1}r_{1}+···+τ_{s}r_{s}} ∥  ∑_{2^{τ_{j} −1}−1≤|m_{j}|<2^{τ_{j}} −1}(a_{j(m)} + ib_{j(m)})e^{2πi(m,x)}∥_{p}.  (9)

Note that in the case of even p = 2N functions (8) and (9) can be directly expressed via f̂(m) = a_{j(m)} + ib_{j(m)}:

∥∑_{m∈Γ} f̂(m)e^{2πi(m,x)}∥_{p}^{p} = ∫_{[0,1]^{s}} |∑_{m∈Γ} f̂(m)e^{2πi(m,x)}|^{2N} dx = (∑_{m∈Γ} |f̂(m)|^{2})^{N}.

The case of p = 2, θ = ∞ was studied in [6].

5. The Korobov and Sobolev classes. Suppose that λ(m) > 0 (m ∈ ℤ^{s}) and the class E_{s}^{θ}(λ) (1 ≤ θ ≤ ∞) consists of summable functions f(x_{1}, …, x_{s}), which are 1-periodic with respect to every variable and are such that

∑_{m∈ℤ^{s}} λ^{−1}(m) < ∞

and

∥λ(m)|f̂(m)|∥_{l^{θ}(ℤ^{s})} ≤ 1.

Let

λ(m) = (∏_{j=1}^{s} max(|m_{j}|, 1))^{r}, r > 1,

and then, for θ = ∞, the set E_{s}^{∞}(λ) ≡ E_{s}^{r} is Korobov’s class [15], and for θ = 2 the set E_{s}^{2}(λ) = W_{2}^{r} is Sobolev’s class with a dominant mixed derivative.

It is clear that in order to specify measures on the classes E_{s}^{θ}(λ), we must use Theorems 1 and 2

for Γ_{m} = {m}, Ψ_{m}(y) = |y|λ(m), if θ = ∞ and Ψ_{m}(y) = (|y|λ(m))^{θ} if 1 ≤ θ < ∞, respectively. Note that on the class E_{s}^{r} the probability measure was introduced in [3], and on the class W_{2}^{r} in [12] (see also [4]).

6. In all cases of measure introduction on classes of functions presented above, we made an essential use of the specification of a function by means of the definition of a complete collection of its trigonometric Fourier coefficients.

We can approach this problem in a different way, namely, via the values of the defining trigonometric polynomials at the nodes of lattices as it was proposed in [7] for the classes H_{p}^{r} ≡ B_{p,∞}^{r}. Namely, (we restrict the consideration to the case p = 2, s = 1) for the 1-periodic function f(x) ∈ L^{2}(0, 1) we set (k = 0, ±1, ±2, …)

f_{k}(x) = ∑_{m∈Γ_{k}} f̂(m)e^{2πi(m,x)},

where Γ_{0} = {−1, 0, 1} and Γ_{k} = {sign k(2^{|k|−1} + 1), …, sign k(2^{|k|})}, for all k ≠ 0.

We refer to the class B_{2,θ}^{r} all functions f(x) such that

∥2^{|k|r}∥f_{k}∥_{2}∥_{l^{θ}(ℤ)} ≤ 1.

It is easy to see that

∥f_{k}∥_{2}^{2} = ∑_{m∈Γ_{k}} |f̂(m)|^{2} = \frac{1}{2^{|k|−1}} ∑_{ν=1}^{2^{|k|−1}}|f_{k}(\frac{ν}{2^{|k|−1}} )|^{2}.

Next, for the given collection of complex numbers {y_{ν}}_{ν∈Γ_{k}} there exists a uniquely defined by it collection of numbers {c_{ν}}_{ν∈Γ_{k}} such that

y_{ν} = ∑_{m∈Γ_{k}} c_{m} exp 2πim\frac{ν}{2^{|k|−1}}

for all ν ∈ Γ_{k}.

Therefore, setting Ψ_{k} = g_{k} for θ = ∞ and Ψ_{k} = g_{k}^{θ} for 1 ≤ θ < ∞, where

g_{k}(x_{1}, …, x_{2^{|k|}}) = 2^{|k|r}(\frac{1}{2^{|k|−1}} ∑_{ν=1}^{2^{|k|−1}}(x_{2ν−1}^{2} + x_{2ν}^{2}))^{1/2},

we obtain, by virtue of Theorems 1 and 2, the probability measure in the classes B_{2,∞}^{r} and B_{2,θ}^{r} respectively.

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Translated by I. Aleksandrova