When Are Borel Sets Continuity Sets

NON-RELATIVISTIC QUANTUM MECHANICS

Michael Dickson , in Philosophy of Physics, 2007

7.5.5 Borel Sets

Borel sets of real numbers are definable as follows. Given some set, S, a σ-algebra over S is a family of subsets of S closed under complement, countable union and countable intersection. The Borel algebra over $ℝ$ is the smallest σ-algebra containing the open sets of $ℝ$ . (One must show that there is indeed a smallest.) A Borel set of real numbers is an element of the Borel algebra over $ℝ$ . Note that not every subset of real numbers is a Borel set, though the ones that are not are somewhat exotic. All open and closed sets are Borel. The importance of Borel algebras (hence Borel sets) lies in the fact that certain measure-theoretic results apply only to them. On the other hand, in many cases one can extend the important results and definitions to a wider class of sets, for example, all sets that are the image of a Borel set under a continuous function. However, we shall not continue to make note of such points.

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Set Theory

Marion Scheepers , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.C The Baire Property

Another notion of smallness emerged with the study of continuity: Nowhere denseness and first category. A set N of reals is nowhere dense if there is for each nonempty open interval U a nonempty open subinterval Vsuch that N is disjoint from V. A set is a first category set if it is a union of countably many nowhere dense sets.

Though first category is a notion of smallness it does not imply small cardinality. One has the following Galilean paradox: The Cantor middle-thirds set has cardinality equal to that of the real line, but it is nowhere dense. Also, first category does not imply measure zero, and vice versa. In fact one has the following Galilean paradox: The real line is a union of two sets A and B where A has Lebesgue measure zero and B has first category.

A set S of real numbers is said to have the Baire property if there area Borel set B and a first category set F such that S  =   (B ⧹ F) ∪ (F ⧹ B). Thus, sets having the Baire property differ only "slightly" from Borel sets in that they differ from a Borel set by a first category set of points.

There are sets of real numbers which do not have the Baire property. The first examples were considered pathological. Moreover, there are many analogies between the notions of Lebesgue measure zero and first category. Do projective sets have the Baire property? Are theorems mentioning Lebesgue measurable/measure zero or Baire property/first category in their hypotheses or conclusions true when Lebesgue measurable and Baire property are interchanged, and Lebesgue measure zero and first category are interchanged?

III.C.1 Projective Sets and the Baire Property

Each Borel set has the Baire property. Lusin and Sierpiński proved that every Σ ₁ ¹ set has the Baire property, and thus every Π₁ ¹ set has the Baire property. Gödel showed that in L there is a set of real numbers which is both Σ₂ ¹ and Π₂ ¹, but does not have the Baire property.

III.C.2 Sierpiński's Duality Program

Sierpiński observed that often a theorem remains true when all occurrences of "Lebesgue measure zero" are replaced with "first category" and vice versa. This observation is summarized by saying that "measure and category are dual notions." Erdös and Sierpiński proved that in every model of set theory in which CH holds the following is true: Let σ be a statement involving only the notions of Lebesgue measure zero, first category, and set theoretic properties, and let σ^* be the statement obtained by interchanging all occurrences of "measure zero" and "first category" in σ. Then σ implies σ^*. The statement in italics is known as the Erdös–Sierpiński duality principle.

Does the duality principle hold in all models of set theory? Rothberger introduced the following combinatorial approach to the problem: Let $\mathcal{N}$ denote the set of all Lebesgue measure zero sets of real numbers, and let $\mathcal{M}$ denote the set of all first category sets of reals. Define the cardinal numbers:

add( $\mathcal{N}$ ) The minimal cardinality of a family of measure zero sets whose union is not measure zero;

cov( $\mathcal{N}$ ) The minimal cardinality of a family of measurezero sets whose union is the real line;

unif( $\mathcal{N}$ ) The minimal cardinality of a set of real numbers which is not of measure zero;

cof( $\mathcal{N}$ ) The minimal cardinality of a family of measure zero sets such that each measure zero set is a subset of some member of this family.

The cardinals add( $\mathcal{M}$ ), cov( $\mathcal{M}$ ), unif( $\mathcal{M}$ ), and cof( $\mathcal{M}$ ) are defined similarly.

Rothberger proved that in any model of set theory both cov( $\mathcal{M}$ )   ≤   unif( $\mathcal{N}$ ) and cov( $\mathcal{N}$ )   ≤   unif( $\mathcal{M}$ ) hold. This is an example of duality: Either statement is obtainable from the other by switching $\mathcal{M}$ and $\mathcal{N}$ .

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Integration

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

15.3.2.5 Borel sets

Definition 15.15

ε is said to be a Borel set if ε can be obtained by a countable number of operations, starting from open sets, each operation consisting of taking unions, intersections, or complements.

The difference between a Borel set and σ-algebra (ring) (15.7) is that Ω in the case of a Borel set must be an open set.

The following facts take place for Borel sets.

Claim 15.3

1.

The collection B of all Borel sets in $ℝ$ ^p is a σ-algebra (ring). In fact, it is the smallest σ-algebra (ring) which contains all open sets, that is, if ε ∈ B then ε ∈ M (μ).

2.

If A ∈ M (μ), there exist Borel sets F and G such that F ⊂ A ⊂ G and

(15.96) $\mu (\mathcal{G}-\mathcal{A})=\mu (\mathcal{A}-ℱ)=0$

This follows from (15.95) if we take ε = 1/n and let n → ∞.

3.

If $\mathcal{A}=ℱ\cup (\mathcal{A}-ℱ)$ one can see that A ∈ M (μ) is the union of a Borel set and a set of measure zero.

4.

Borel sets are μ-measurable for every μ (for details see below), but the sets of measure zero (that is, the sets ε for which μ^* (ε) = 0) may be different for different μ's.

5.

For every μ the sets of measure zero from σ-algebra (ring).

6.

In the case of the Lebesgue measure (μ = m) every countable set has measure zero. But there are uncountable (in fact, perfect) sets of measure zero (see Rudin (1976) Chapter 11 with the Cantor set as an example).

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The Internet of Things

Christofer Larsson , in 5G Networks, 2018

Other Model Types

Baccelli and Zuyev use stochastic geometry to describe user densities. Users are assumed to be distributed according to a Poisson point process , where the number of points in a bounded Borel set $\mathcal{B}$ has a Poisson distribution with mean $\mathrm{\Lambda }(\mathcal{B})$ and the numbers of points in disjoint Borel sets are independent random variables. A point process is made a marked point process by attaching a characteristic (mark) to each point of the process, such as active in a call and inactive. The authors also use a stochastic road pattern generated by a Poisson line process and an assumed known velocity distribution of the users on this road pattern to derive expectation and variance of the number of active users per unit area.

The gravity model has been used to model traffic routing and user mobility [133]. The model formulated for traffic flow between two regions is

$T_{i,j}=\frac{m_i{m}_j{P}_i{P}_j}{d_{i,j}^{{\gamma }_i+{\gamma }_j}},$

where $T_{i,j}$ is the traffic from region i to region j, $P_i$ and $P_j$ are the populations of the two regions, and $d_{i,j}$ is their distance (measured between suitable points). The constants $m_i$ , $m_j$ , ${\gamma }_i$ , and ${\gamma }_j$ have to be determined from empirical data. This model can be used for user mobility modeling as well. In fact, the gravity model is incorporated in the proposed mobility model in Section 14.4.

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Random Variables

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques, Volume 2, 2009

2.1.1 Measurable functions

Let now (Ω, $\mathcal{F}$ ) be a measurable space, and (ℝ, ℬ(ℝ)) be a real line with the system ℬ(ℝ) of Borel sets. The following definition is the central one in this section.

Definition 2.1. A real function ξ   =   ξ (ω) defined on (Ω, $\mathcal{F}$ ) is said to be an $\mathcal{F}$ -measurable (or Borel measurable) function or random variable if the following inclusion holds:

(2.1) $\left(\omega :\xi \left(\omega \right)\in B\right)\in \mathrm{ℱ}$

for each set B ϵ ℬ(ℝ) or, equivalently, if the inverse image is a measurable set in Ω, i.e.

(2.2) ${\xi }^{-1}\left(B\right):=\left(\omega :\xi \left(\omega \right)\in B\right)\in \mathrm{ℱ}$

Remark 2.1. Fig. 2.1 illustrates the main properties of usual functions, which state correspondence between each point in ℝ (argument) and some point of ℝ (value function), and an $\mathcal{F}$ -measurable function, which state correspondence between each set B of possible values of function in ℝ and some set B of corresponding realizations ω ('a random factor') from   ⊗   .

Fig. 2.1. Usual and $\mathcal{F}$ -measurable functions.

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Convex Functions, Partial Orderings, and Statistical Applications

In Mathematics in Science and Engineering, 1992

15.14 Theorem

Let g _i (x, y) be a nonnegative AI function on $ℝ$ ⁿ × $ℝ$ ⁿ , i = 1, 2. Then g(x, y) ≡ g ₁(x, y)g ₂(x, y) is AI on $ℝ$ ⁿ × $ℝ$ ⁿ .

The proof is obvious and thus omitted.

A similar preservation under products property holds for AI functions f(π) on S and AI functions h(x) on $ℝ$ ⁿ .

To present the next preservation property of AI functions, we need some definitions.

Let Λ and T be semigroups in $ℝ$ . Let μ be a measure on T. It is said to be invariant if

$\mu (A\cap T)=\mu ((A+x)\cap T)$

for each Borel set A of $ℝ$ and each x ∈T. A measurable function ϕ(λ, x) integrable with respect to μ, defined on Λ ⁿ × T ⁿ is said to have the semigroup property with respect to μ if for each λ₁, λ₂ in Λ ⁿ and x in T ⁿ , ϕ(λ₁ + λ₂, x) = ∫ _{T ⁿ} ϕ(λ₁, x − y) ϕ(λ₂, y) dμ(y ₁) ··· dμ(y _n ).

The next theorem shows that the Schur-convex (Schur-concave) property of functions is preserved under an integral transform by an AI function possessing the semigroup property.

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Markov models and MCMC algorithms in image processing

Xavier Descombes , in Academic Press Library in Signal Processing, Volume 6, 2018

8.4.1 Modeling

Consider K, a compact subset of ${\mathcal{R}}^n$ . K represents the image support, i.e., the image coordinates are embedded in a continuous space. A configuration of points, denoted by x, is a finite unordered set of points in K, such as {x ₁, …, x _n }. The configuration space, denoted by Ω, is therefore written as:

(8.40) $\begin{array}{l}\hfill \mathrm{\Omega }=\bigcup _{n\in \mathcal{N}}{\mathrm{\Omega }}_n,\end{array}$

where Ω₀ = {ø} and Ω _n = {{x ₁, …, x _n }, x _i ∈ K, ∀i} is the set of the configurations of n unordered points for n ≠0. For every Borel set A in K, let N _X (A) be the number of points of X that fall in the set A. A point process is then defined as follows:

Definition 8.3

X is a point process on K if and only if, for every Borel set A in K, N _X (A) is a random variable that is almost surely finite.

Definition 8.4

A point process X on K is called a Poisson process with intensity measure ν(⋅) if and only if:

•

N _X (A) follows a discrete Poisson distribution with expectation ν(A) for every bounded Borel set A in K,

•

for k nonintersecting Borel sets A ₁, A ₂, …, A _k , the corresponding random variables N _X (A ₁), N _X (A ₂), …, N _X (A _k ) are independent.

For every Borel set, B, the probability measure π _ν (B), associated with the Poisson process, is given by [30]:

(8.41) $\begin{array}{l}\hfill {\pi }_{\nu }(B)=e^{-\nu (K)}\left(1_{[\varnothing \in B]}+\sum _{n=1}^{\infty }\frac{{\pi }_{{\nu }_n}(B)}{n!}\right),\end{array}$

with:

(8.42) $\begin{array}{l}\hfill {\pi }_{{\nu }_n}(B)={\int }_K\cdots {\int }_K{1}_{[\{x_1,\dots ,x_n\}\in B_n]}\nu (d{x}_1)\cdots \nu (d{x}_n),\end{array}$

where B _n is the subset of configurations in B which contain exactly n points.

We now define more general processes by considering a density with respect to the Poisson measure. Let f be a probability density with respect to the π _ν (⋅) law of the Poisson process, such that:

(8.43) $\begin{array}{l}\hfill f:\mathrm{\Omega }\to [0,\infty ),{\int }_{\mathrm{\Omega }}f(\mathbf{x})d{\pi }_{\nu }(\mathbf{x})=1.\end{array}$

The measure defined by $P(A)={\int }_{A}f(\mathbf{x})d{\pi }_{\nu }(\mathbf{x})$ , for every Borel set A in Ω, is a probability measure on Ω that defines a point process.

Such a model can favor or penalize geometric properties such as clustering effect or points' alignment, which leads to interesting possibilities for modeling the scene under study. Similarly to MRFs, we introduce the following Markov Property:

Definition 8.5

Let X be a point process with density f. X is a Markov process under the symmetric and reflexive relation ∼ if and only if, for every configuration x in Ω such that f(x) > 0, X satisfies:

•

f(y) > 0 for every y included in x (heredity),

•

for every point u from $K,f(\mathbf{x}\cup \{u\})/f(\mathbf{x})$ only depends on u and its neighborhood $\partial (\{u\})\cap \mathbf{x}=\{x\in \mathbf{x}:u\sim x\}$ (Markov property).

A result similar to the Hammersley-Clifford theorem allows the density of a Markov point process to be decomposed as the product of local functions defined on cliques:

Theorem 8.1

A density which is associated with a point process $f:\mathrm{\Omega }\to [0,\infty [$ is Markovian under the neighborhood relation ∼ if and only if there exists a measurable function $\varphi :\mathrm{\Omega }\to [0,\infty [$ such that:

(8.44) $\begin{array}{l}\hfill \forall \mathbf{x}\in \mathrm{\Omega },f(\mathbf{x})=\alpha \prod _{\mathbf{y}\subseteq \mathbf{x},\mathbf{y}\in {\mathcal{C}}_{\mathbf{x}}}\varphi (\mathbf{y}),\end{array}$

where the set of cliques is given by ${\mathcal{C}}_{\mathbf{x}}=\{\mathbf{y}\subseteq \mathbf{x}:\forall \{u,v\}\subseteq \mathbf{y},u\sim v\}$ .

Just as for the random field case, we can then write the density in the form of a Gibbs density:

(8.45) $\begin{array}{l}\hfill f(\mathbf{x})=\frac{1}{c}exp\left[-\sum _{\mathbf{y}\subseteq \mathbf{x},\mathbf{y}\in {\mathcal{C}}_{\mathbf{x}}}V(y)\right],\end{array}$

where $U(\mathbf{x})={\sum }_{\mathbf{y}\subseteq \mathbf{x},\mathbf{y}\in {\mathcal{C}}_{\mathbf{x}}}V(\mathbf{y})$ is called the energy and V (y) is a potential.

A typical example of a Markov/Gibbs process, which is often used, is the pairwise interaction process. In this case, a neighborhood relation ∼ is defined between pairs of points. For example, x _i ∼ x _j if and only if d(x _i , x _j ) < r, where d(⋅, ⋅) is the Euclidean distance and r is a radius of interaction. In this case, the unnormalized density is written as:

(8.46) $\begin{array}{l}\hfill h(\mathbf{x})=\prod _{i=1}^{n(\mathbf{x})}b(x_i)\prod _{1\le i<j\le n(\mathbf{x});x_i\sim x_j}g(x_i,x_j),\end{array}$

where n(x) is the number of points in the configuration x.

In image analysis applications, the goal is to extract a set of objects rather than a set of points. We therefore associate a low dimensional parametric object to each point. A marked point is then defined by its location x _i and a random vector m _i ∈ M, called the mark, which defines the underlying object (e.g., the radius in the case of a circle). We then have the following definition:

Definition 8.6

A marked point process on χ = K × M is a point process on χ for which the positions of the points are in K and the marks are in M, such that the unmarked point process is a well defined point process on K.

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Geometric Function Theory

Ch. Pommerenke , in Handbook of Complex Analysis, 2002

5.1 Subdomains of the unit disk

A common situation is that one has domains G and H with G ⊂ H and is interested in ∂G ⋂ ∂H. For instance G might be a domain under investigation whereas H is a simpler domain with known properties. Let g and h be conformal maps of $\mathbb{D}$ onto G and H respectively. Then f = h ⁻¹ ∘ g maps $\mathbb{D}$ conformally onto $F=h^{-1}(G)\subset \mathbb{D}$ and there is a set A with

(5.1.1) $\begin{array}{ll}A\subset \mathbb{T},\hfill & f(A)\subset \mathbb{T}\hfill \end{array}$

such that g(A) = ∂G ⋂ ∂H in a suitable sense; see Figure 13.

Fig. 13. How to reduce the case G ⊂ H to $F\subset \mathbb{D}$ .

For $\zeta \in \mathbb{T}$ let f(ζ) denote the angular limit whenever it exists; it can fail to exist only on a set of zero capacity and thus of zero measure (Theorem 2.9). It is no essential restriction to assume that f(0) = 0.

Theorem 5.1

Let f map $\mathbb{D}$ conformally into $\mathbb{D}$ and let $A\subset \mathbb{T}$ be a Borel set with $f(A)\subset \mathbb{T}$ . If f(0) = 0 then

(5.1.2) $\Lambda \left(f(A)\right)\ge \Lambda (A),$

(5.1.3) $\text{cap}f(A)\ge \frac{\text{cap}A}{\sqrt{\left|f^{\prime }(0)\right|}}\ge \text{cap}A.$

Thus f increases the size of sets A on $\mathbb{T}$ provided that f(A) also lies on $\mathbb{T}$ . The estimate (5.1.2) is Löwner's Lemma and is valid also if f is not injective. Its invariant form is Carleman's principle of domain extension for harmonic measure [92, p. 68]. See [105, p. 217] for (5.1.3).

These results were extended to Hausdorff measures by Makarov [76], [105, p. 234] and to generalized capacities by Hamilton [43]. Both results imply that

(5.1.4) $\begin{array}{ll}\text{dim}f(A)\ge \text{dim}A\hfill & \text{if}A\subset \mathbb{T}\hfill \end{array},f(A)\subset \mathbb{T},$

where dim is the Hausdorff dimension [29].

Theorem 5.2

Let f map $\mathbb{D}$ conformally into $\mathbb{D}$ . If f(ζ) exists and lies on $\mathbb{T}$ then the angular derivative f′(ζ) exists and

(5.1.5) $0<\left|f^{\prime }(\zeta )\right|\le +\infty .$

Moreover f′(ζ) ≠ ∞ for almost all ζ with $f(\zeta )\in \mathbb{T}$ .

The first part follows from the Julia–Wolff Lemma [105, p. 82] valid for all analytic selfmaps of $\mathbb{D}$ . The second part [82] follows at once from Theorem 4.9 because $f:\mathbb{D}\to \mathbb{D}$ cannot be twisting at any ζ with $f(\zeta )\in \mathbb{T}$ .

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SET THEORY FROM CANTOR TO COHEN

Akihiro Kanamori , in Philosophy of Mathematics, 2009

2.5 Analytic and Projective Sets

A decade after Lebesgue's seminal paper [1905], descriptive set theory emerged as a distinct discipline through the efforts of the Russian mathematician Nikolai Luzin. He had become acquainted with the work of the French analysts while in Paris as a student and had addressed Baire's functions with a intriguing use of CH. What is now known as a Luzin set is an uncountable set of reals whose intersection with any meager set is countable, and Luzin established: CH implies that there is a Luzin set. ⁶⁷ This would become a paradigmatic use of CH, in that a recursive construction was carried out in ${\aleph }_1$ steps where at each state only countable many conditions have to be attended to, in this case by applying the Baire Category Theorem. Luzin showed that the characteristic function of his set escaped Baire's function classification, and Luzin sets have since become pivotal examples of "special sets" of reals.

In Moscow Luzin began an important seminar, and from the beginning a major topic was the "descriptive theory of functions". The young Pole Wacł Sierpiński was an early participant while he was interned in Moscow in 1915, and undoubtedly this not only kindled a decade-long collaboration between Luzin and Sierpiński but also encouraged the latter's involvement in the development of a Polish school of mathematics and its interest in descriptive set theory.

Of the three regularity properties, Lebesgue measurability, the Baire property, and the perfect set property (cf. 2.3 ), the first two were immediate for the Borel sets. However, nothing had been known about the perfect set property beyond Cantor's own result that the closed sets have it and Bernstein's that with a well-ordering of the reals there is a set not having the property. Luzin's student Pavel Aleksandrov [1916] established the groundbreaking result that the Borel sets have the perfect set property, so that "CH holds for the Borel sets". ⁶⁸

In the work that really began descriptive set theory another student of Luzin's, Mikhail Suslin, investigated the analytic sets following a mistake he had found in Lebesgue's paper. ⁶⁹ Suslin [1917] formulated these sets in terms of an explicit operation ⁷⁰ and announced two fundamental results: a set B of reals is Borel iff both B and ℝ ‐ B are analytic; and there is an analytic set which is not Borel. ⁷¹ This was to be his sole publication, for he succumbed to typhus in a Moscow epidemic in 1919 at the age of 25. In an accompanying note Luzin [1917] announced the regularity properties: Every analytic set is Lebesgue measurable, has the Baire property, and has the perfect set property, the last result attributed to Suslin.

Luzin and Sierpiński in their [1918] and [1923] provided proofs, and the latter paper was instrumental in shifting the emphasis toward the co-analytic sets, i.e. sets of reals X such that ℝ – X is analytic. They used well-founded relations to provide a basic tree representation of co-analytic sets, one from which the main results of the period flowed, and it is here that well-founded relations entered mathematical practice. ⁷²

After the first wave in descriptive set theory brought about by Suslin [1917] and Luzin [1917] had crested, Luzin [1925a] and Sierpiński [1925] extended the domain of study to the projective sets. For Y ⊆ ℝ _k+1 and with ordered k-tuples defined from the ordered pair, the projection of Y is

$pY=\left\{〈x_1,\dots ,x_k〉\right|\exists y(〈x_1,\dots ,x_k,y〉\in Y)\}.$

Suslin [1917] had essentially noted that a set of reals is analytic iff it is the projection of a Borel subset of ℝ². ⁷³ Luzin and Sierpiński took the geometric operation of projection to be basic and defined the projective sets as those sets obtainable from the Borel sets by the iterated applications of projection and complementation. The corresponding hierarchy of projective subsets of ℝ^k is defined, in modern notation, as follows: For $A\subseteq {ℝ}^k$ ,

$A\text{is}{\displaystyle \sum {}_1^1}iffA=pY\text{for\hspace{0.17em}some\hspace{0.17em}Borel\hspace{0.17em}set}Y\subseteq {ℝ}^{k+1},$

i.e. A is analytic ⁷⁴ and for integers n > 0,

$\begin{array}{c}A\text{is}{\displaystyle \prod {}_n^1}iff{ℝ}^k-A\text{is}{\displaystyle \Sigma {}_n^1},\\ A\text{is}{\displaystyle \Sigma {}_{n+1}^1}iffA=pY\text{for\hspace{0.17em}some}{\displaystyle \prod {}_n^1}\text{\hspace{0.17em}set}Y\subseteq {ℝ}^{k+1},\text{and}\\ A\text{is}{\Delta }_n^{1}iffA\text{is}\text{both}{\displaystyle \Sigma {}_n^1{\displaystyle \prod {}_n^1}}.\end{array}$

Luzin [1925a] and Sierpiński [1925] recast Lebesgue's use of the Cantor diagonal argument to show that the projective hierarchy is proper, and soon its basic properties were established. However, this investigation encountered basic obstacles from the beginning. Luzin [1925a] emphasized that whether the ${\displaystyle \prod {}_1^1}$ sets, the co-analytic sets at the bottom of the hierarchy, have the perfect set property was a major question. In a confident and remarkably prophetic passage he declared that his efforts towards its resolution led him to a conclusion "totally unexpected", that "one does not know and one will never know "of the family of projective sets, although it has cardinality $2^{{\aleph }_0}$ and consists of "effective sets", whether every member has cardinality $2^{{\aleph }_0}$ if uncountable, has the Baire property, or is even Lebesgue measurable. Luzin [1925b] pointed out the specific problem of establishing whether the sets are Lebesgue measurable. Both these difficulties were also pointed out by Sierpiński [1925]. This basic impasse in descriptive set theory was to remain for over a decade, to be surprisingly resolved by penetrating work of Gödel involving metamathematical methods (cf. 3.4).

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When risks and uncertainties collide: quantum mechanical formulation of mathematical finance for arbitrage markets

Simone Farinelli , Hideyuki Takada , in Mechanics and Physics of Structured Media, 2022

Appendix 16.A Generalized derivatives of stochastic processes

In stochastic differential geometry one would like to lift the constructions of stochastic analysis from open subsets of ${\mathbb{R}}^N$ to N dimensional differentiable manifolds. To that aim, chart invariant definitions are needed and hence a stochastic calculus satisfying the usual chain rule, and not Itô's Lemma is required, (cf. [14], Chapter 7, and the remark in Chapter 4 at the beginning of page 200). That is why the papers about geometric arbitrage theory are mainly concerned in by stochastic integrals and derivatives meant in Stratonovich's sense and not in Itô's. Of course, at the end of the computation, Stratonovich integrals can be transformed into Itô's. Note that a fundamental portfolio equation, the self-financing condition cannot be directly formally expressed with Stratonovich integrals, but first with Itô's and then transformed into Stratonovich's, because it is a nonanticipative condition.

Definition 16.A.1

Let I be a real interval and $Q={(Q_t)}_{t\in I}$ be a ${\mathbb{R}}^N$ -valued stochastic process on the probability space $(\mathrm{\Omega },\mathcal{A},P)$ . The process Q determines three families of σ-subalgebras of the σ-algebra $\mathcal{A}$ :

(i)

"Past" ${\mathcal{P}}_t$ , generated by the preimages of Borel sets in ${\mathbb{R}}^N$ by all mappings $Q_s:\mathrm{\Omega }\to {\mathbf{R}}^N$ for $0<s<t$ .

(ii)

"Future" ${\mathcal{F}}_t$ , generated by the preimages of Borel sets in ${\mathbb{R}}^N$ by all mappings $Q_s:\mathrm{\Omega }\to {\mathbf{R}}^N$ for $0<t<s$ .

(iii)

"Present" ${\mathcal{N}}_t$ , generated by the preimages of Borel sets in ${\mathbb{R}}^N$ by the mapping $Q_s:\mathrm{\Omega }\to {\mathbf{R}}^N$ .

Let $Q={(Q_t)}_{t\in I}$ be continuous. Assuming that the following limits exist, Nelson's stochastic derivatives are defined as

(16.A.1)

Let ${\mathcal{S}}^1(I)$ the set of all processes Q such that $t\mapsto Q_t$ , $t\mapsto \mathfrak{D}{Q}_t$ , and $t\mapsto {\mathfrak{D}}_{⁎}{Q}_t$ are continuous mappings from I to $L^2(\mathrm{\Omega },\mathcal{A})$ . Let ${\mathcal{C}}^1(I)$ the completion of ${\mathcal{S}}^1(I)$ with respect to the norm

(16.A.2)

Remark 16.A.2

The stochastic derivatives $\mathfrak{D}$ , ${\mathfrak{D}}_{⁎}$ , and $\mathcal{D}$ correspond to Itô's, to the anticipative, and, respectively, to Stratonovich's integral (cf. [13]). The process space ${\mathcal{C}}^1(I)$ contains all Itô processes. If Q is a Markov process, then the sigma algebras ${\mathcal{P}}_t$ ("past") and ${\mathcal{F}}_t$ ("future") in the definitions of forward and backward derivatives can be substituted by the sigma algebra ${\mathcal{N}}_t$ ("present"), see Chapters 6.1 and 8.1 in [13].

Stochastic derivatives can be defined pointwise in $\omega \in \mathrm{\Omega }$ outside the class ${\mathcal{C}}^1$ in terms of generalized functions.

Definition 16.A.3

Let $Q:I\times \mathrm{\Omega }\to {\mathbb{R}}^N$ be a continuous linear functional in the test processes $\phi :I\times \mathrm{\Omega }\to {\mathbb{R}}^N$ for $\phi (\cdot ,\omega )\in C_c^{\infty }(I,{\mathbb{R}}^N)$ . We mean by this that for a fixed $\omega \in \mathrm{\Omega }$ the functional $Q(\cdot ,\omega )\in \mathcal{D}(I,{\mathbb{R}}^N)$ , the topological vector space of continuous distributions. We can then define Nelson's generalized stochastic derivatives:

(16.A.3)

If the generalized derivative is regular, then the process has a derivative in the classic sense. This construction is nothing else than a straightforward pathwise lift of the theory of generalized functions to a wider class stochastic processes which do not a priori allow for Nelson's derivatives in the strong sense. We will utilize this feature in the treatment of credit risk, where many processes with jumps occur.

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Source: https://www.sciencedirect.com/topics/engineering/borel-set

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