An element set labelling a Cartesian product by measurable binary relations which leads to postulates of the theory of experience and chance as a theory of co~events

An element-set labelling a Cartesian

product by measurable binary relations

which leads to postulates of the theory of

experience and chance as a theory of

co events

Vorobyev, Oleg Yu.

Siberian Federal University, Institute of Mathematics and Computer

Science

30 September 2016

Online at

https://mpra.ub.uni-muenchen.de/81891/

An element-set labelling a Cartesian product by measurable

binary relations which leads to postulates of the theory of

experience and chance as a theory of co

∼

events

Oleg Yu. Vorobyev

Institute of mathematics and computer science Siberian Federal University

Krasnoyarsk

mailto:[email protected] http://www.sfu-kras.academia.edu/OlegVorobyev

http://olegvorobyev.academia.edu

Abstract. We introduce the set-theoretic language for the element-set labelling a Cartesian product by measurable binary relations intended for the labelling, or for the naming of parts and details of the construction that we are going to propose in the theory of experience and chance, or the theory of co∼events that serve as mathematical models of events as dual pairs.

Keywords.Eventology, theory of experience and chance, theory of co∼events, measurable binary relation, event, co∼event, experience, chance.

Before you start working in a set-theoretic space whose objects of interest are simultaneously the

elements of space, the sets of elements, and the sets of subsets of elements, it is necessary to stock up some system of «coordinates» suitable for a labelling the space itself, and its parts.

1 A labelling set and a set of its labelling subsets

Here, in my opinion, a slightly peculiar but effective system of set-theoretic coordinates «elements—sets» is quite suitable, based on some labelling setXand some set S X

⊆P₍X₎of its labelling subsets, and also on theM-complement1_X(c)_{of the labelling setX, and on the one-to-one corresponding to the set S} X

the set of its labelling subsets S X(c) ={Xc(c):X ∈ S X}⊆P(_X(c))

.

Warning 1(relative subsets and relative empty subsets). Since in the theory of experience and chance

one has to deal simultaneously with subsets of sets of different levels, we will need unusual, but convenient notation, directly indicating what subsets of which set is spoken. For example, if we are talking about subsetsx⊆Ω,X⊆X_{, or}O_⊆P₍X_{), then denotations of subsetsx, X, or}O_{, when appropriate,}

we will write more fully:x//Ω,X//X_{, or} O_//P₍X_{), directly specifying in which sets these subsets contain.}

Especially we will have to deal with empty subsets:∅_//Ω,∅//X_{, or∅//}P₍X_{), for which we introduce more}

compact notation:∅Ω₌ ∅_//Ω,∅X ₌∅_//X_{, or}∅P(X₎

=∅//P₍X_{), we will talk about them asrelatively empty}

subsets, and callΩ-empty,X-empty, orP₍X)-empty subsetscorrespondingly.

Consider themeasurable space(Ω,A)composed of some setΩand a sigma-algebraAof its subsets and we emphasize that:elementsω ∈Ω;measurable subsetsx⊆Ω;some setX₌{x:x∈ A} ⊆ A, composed from measurable subsetsx∈X_{; andsome set S} X_⊆P₍X_{)of subsetsX ⊆}X_{, consisting from measurable subsets} x ∈ X ⊆ X_{;until they have no meaningful interpretationand form only a basis Λpeculiar element-set}

labels_{λ∈Λ(tags, dockets, tickets, ornames), intended for a element-set labelling, or a nominating the}

parts and details of the construction that we are going to propose in the theory of experience and chance [5, 4] as a mathematical model of an event as a dual pair.

Predefinition 1(Basic element-set labels). Basic element-set labels λ ∈ Λ are called as elements,

sets and sets of subsets of the measurable space (Ω,A), and also results of terraced set-theoretic

c

⃝ 2016 O.Yu.Vorobyev

Oleg Vorobyev (ed.), Proc. XV FAMEMS’2016, Krasnoyarsk: SFU

operationsover them, equipped with their own titles, a list of which can be found in the Appendix on page 122.

We’llfill up the stock ofΛtags with one more label, Cartesian product

X× S X

={(x, X) :x∈X_{, X}∈ S X}, (1)

which defines a binary relation

R_X,SX =

{

(x, X) :x∈X, x∈X_{, X ∈ S} X}_⊆X_{× S} X ₍₂₎

as amembership relationx∈X between elementsx∈X_{and subsetsX ∈ S} X_{; and also a complementary}

binary relation

Rc_X

,SX =

{

(x, X) :x̸∈X, x∈X_{, X ∈ S} X}_⊆X_{× S} X ₍₃₎

as anon-membership relationx̸∈Xbetween elementsx∈Xand subsetsX∈ S X

; so that

R_X

,SX+Rc

X,SX =X× S

X

. (4)

Finally, we add to the stockΛso calledterraced2_label

(

TerX//X, ter(X//X) )

=

( ∪

x∈X

x, ∩

x∈X

x ∩

x∈X−X

(Ω−x)

)

⊆Ω×Ω, (5)

numbered by labels-subsetsX ∈ S Xand while defined simply as a pair of indicated measurable subsets ofΩ.

To have a full stock we’ll stock up in the literal sense «complementary» element-set labels, constructed from: 1) the complements xc _{= Ω−xto measurable subsets x ⊆ Ω, 2) the}

М-complementary setX(c) ₌

{xc_{:x∈ X} ⊆ Acomposed from these complements, and 3) the sets S} X(c)

={Xc(c)_{:X ∈ S} X}

⊆ P(_X(c)) of

subsetsXc(c)_{= (X}c₎(c)_{= (X−X)}(c)_⊆X(c)_{, i.e., such thatX}c(c)_={xc_{: x∈X}c_{} ∈ S} X(c) .

There we also place a label similar to (11), the Cartesian product

X(c)× S X(c)

={(xc, Xc(c)) :xc∈X(c)_{, X}c(c)∈ S X(c)}

, (6)

which defines analogous to (13) a complementary binary relation

Rc

X(c)_,SX(c) =

{

(xc, Xc(c)) :xc∈Xc(c), xc∈X(c)_{, X}c(c)_{∈ S} X(c)}_⊆X(c)_{× S} X(c) ₍₇₎

as a membership relation xc _{∈ X}c(c) _{between elements x}c _{∈ X}(c) _{and subsets X}c(c) _{∈ S} X(c)

; and also a complementary binary relation

R

X(c)_,SX(c) =

{

(xc, Xc(c)) :xc̸∈Xc(c), xc∈X(c)_{, X}c(c)_{∈ S} X(c)}_⊆X(c)_{× S} X(c) ₍₈₎

as anon-membership relationxc _̸∈Xc(c)_{between elementsx}c _∈X(c)_{and subsetsX}c(c)_{∈ S} X(c)

; so that

Rc

X(c)_{, S}X(c)+RX(c)_{, S}X(c) =X

(c)_{× S} X(c)

. (9)

2_{Those who are familiar with the beginnings of the eventological theory [3, 2007] should keep their attention to the amazing}

inevitability of the «splitting» of the previously unified concept of theterrace eventinto two dual halves, the right of which is the

terraced ket-eventwhich is defined as a terrace event of thefirst kindter(X//X) = ∩

x∈X x ∩

x∈X−_X

(Ω−x)⊆Ωfrom the eventological

part of the Kolmogorovprobability theory, and the left one is aterraced bra-event, a new concept from thetheory of believabilities,

dual to theprobability theory, which is defined as terraced event of the 5th kindTerX//X =

∪

x∈X

x ⊆ Ωfrom the eventological

Finally, do not forget the similar to (15)terrace label

(

TerXc(c)_//X(c), ter

(

Xc(c)_//X(c)) )₌



 ∪

xc_∈Xc(c)

xc_, ∩

xc_∈Xc(c)

xc ∩

xc_∈X(c)_−Xc(c)

(Ω−xc₎



⊆Ω×Ω, (10)

numbered by labels-subsetsXc(c)_{∈ S} X(c) .

Warning 2(membership relations and paradoxes of naive set theory). Some mathematical relations such

as «member of» and «subset of», generally speaking, should not be understood as binary relations because its domains and codomains cannot be sets in usual systems of axiomatic set theory. For example, if you try to model the general concept of membership as a binary relation «∈», then then for this you will have to define the domain and the codomain, which can be a class of all sets. But such a class is not a set in the naive set theory, and the assumption that the relation «∈» is defined on all sets leads to a contradiction from the well-known Russell paradox. At the same time, in the overwhelming majority of mathematical contexts, links to the relation «member of» and «subset of» are absolutely harmless, because they are tacitly limited to some set which is clear from the context. The removal of this problem consists in choosing each time a sufficiently large setA, which contains all objects of interest, and work with the restriction «∈A» instead of «∈». Similarly, the relation «⊆» must also be limited to the relation

«⊆A» to have some domainAand the codomainP(A), set of all subsets ofA. Therefore, the chain of three

membership relations

ω ∈ x ∈ X ∈ S X ⊆ P₍X_{) (11)}

will always be understood by me as

ω ∈Ω x ∈X X ∈ SX S

X

⊆P(X₎ P(X), (12)

the chain oflimited by defaultmembership relations.

The stock Λ of element-set labels λ ∈ Λ is intended to construct such a system of element-set «coordinates», which, relying on a duality «element–_{set», will allow us to divide each concept of the} theory of experience and chance (TEC)into two dual parts and present it in the form of a conveniently writtendual pair, i.e., pairs composed of two dual parts. In the bra-ket notation, the dual parts of pairs labelled with the labelsλ, λ′ _∈

Λ, are denoted by⟨λ| and |λ′_{⟩correspondingly, the entire dual pair is} denoted by⟨λ|λ′_{⟩and is defined as the Cartesian product⟨λ|λ}′_{⟩=⟨λ| × |λ}′_{⟩of their dual parts, placing the} corresponding concept of thetheory of experience and chancein the system of element-set «coordinates».

2 Binary relations and quotient-sets

Definition 1(Cartesian product of measurable spaces). Let⟨Ω,A| = (⟨Ω|,⟨A|) be the measurable

bra-space, and|Ω,A⟩= (|Ω⟩,|A⟩)be the measurable ket-space. Let’s denote

⟨Ω|Ω⟩=⟨Ω| × |Ω⟩={

⟨ω∗|ω⟩ : ⟨ω∗| ∈ ⟨Ω|,|ω⟩ ∈ |Ω⟩}

the Cartesian product of sets⟨Ω|and|Ω⟩called abra-ket-set. The Cartesian product of the sigma-algebra ⟨A| × |A⟩is a family of subsets of⟨Ω| × |Ω⟩. In general, this family is not closed with respect to countable unions, and hence is not a sigma-algebra. We shall introduce the notation

⟨A|A⟩=σ(⟨A| × |A⟩)

for the minimal sigma-algebraσ(⟨A| × |A⟩)containing ⟨A| × |A⟩, and we shall call it the bra-ket sigma-algebra. Then the pair of the bra-ket-set and the bra-ket sigma-algebra, i.e., the measurable bra-ket-space

⟨Ω,A|Ω,A⟩= (⟨Ω|Ω⟩,⟨A|A⟩),

is called the(Cartesian) product of measurable spaces⟨Ω,A|and|Ω,A⟩.

Definition 2(cross-sections of a measurable binary relation). Let R _{⊆ ⟨Ω|Ω⟩ be some measurable}

binary relationon⟨Ω|Ω⟩. Let’s denote

thecross-section ofR_{by the ket-point|ω⟩ ∈ |Ω⟩which by definition serves as a measurable subset of the}

bra-set⟨Ω|, i.e.,R_{||ω⟩∈ ⟨}A_{|, and}

R_|⟨ω∗_|={|ω⟩ ∈ |Ω⟩ | ⟨ω∗|ω⟩ ∈R} ⊆ |Ω⟩

thecross-section ofR_{by the bra-point⟨ω}∗_{| ∈ ⟨Ω|which by definition serves as a measurable subset of the} ket-set|Ω⟩, i.e.,R_|⟨ω∗_|∈ |A⟩.

Complementary to theR_{a binary relationhas a standard denotation}Rc_{=⟨Ω|Ω⟩ −}R_{, and its cross-sections}

have the form

Rc_{||ω⟩={⟨ω}∗_{| ∈ ⟨Ω| | ⟨ω}∗_{|ω⟩ ∈}Rc_{} ⊆ ⟨Ω|,}

thecross-section ofRc_{by the ket-point|ω⟩ ∈ |Ω⟩, which by definition serves as a measurable subsetof the}

bra-set⟨Ω|, i.e.,Rc_|

|ω⟩∈ ⟨A|, and

Rc_|⟨ω∗_|={|ω⟩ ∈ |Ω⟩ | ⟨ω∗|ω⟩ ∈Rc} ⊆ |Ω⟩,

thecross-section ofRc_{by the bra-point⟨ω}∗_{| ∈ ⟨Ω|, which by definition serves as a measurable subset of the} ket-set|Ω⟩, i.e.Rc_|

⟨ω∗_|∈ |A⟩.

Property 1(duality of cross-sections).The following membership relations are equivalence:

⟨ω∗_{| ∈R|}

|ω⟩ ⇐⇒ |ω⟩ ∈R|⟨ω∗_|. (13)

Proofis obvious, since the left and the right membership relations in (13) are equivalence to the same

membership relation⟨ω∗_{|ω⟩ ∈R, which follows from Definition 2.}

Note 1(cross-sections of complementary binary relations). From the definition of cross-sections of

complementary binary relationsR_andRc _{it follows that cross-sections by⟨ω}∗_{|and by|ω⟩are mutually}

complementary in the ket-set |Ω⟩ and in the bra-set ⟨Ω| correspondingly: R_|⟨ω∗_| +Rc|⟨_ω∗_| = |Ω⟩ and R_||ω⟩+Rc_{||ω⟩=⟨Ω|.}

2.1 Bra-relation and ket-relation of equivalence, generated by a binary relation

Definition 3(two equivalence relations, generated by a binary relation). Each binary relation

R_{⊆ ⟨Ω|Ω⟩deines two equivalence relations: thebra-relation of equivalenceon⟨Ω|:}

⟨R_|={

⟨ω∗, ω∗′|:R_|⟨ω∗_|=R|⟨_ω∗′_|}⊆ ⟨Ω| × ⟨Ω|, (14)

in other words, for⟨ω∗_{, ω}∗′_{| ∈ ⟨Ω| × ⟨Ω|}

⟨ω∗| ∼⟨R|⟨ω∗| ⇐⇒R|⟨ω∗_|=R|⟨_ω∗_|; (15)

and theket-relation of equivalenceon|Ω⟩:

|R_⟩={_{|ω, ω}′_⟩:R_||ω⟩=R_||ω′_⟩}⊆ |Ω⟩ × |Ω⟩, (16)

in other words, for|ω, ω′_{⟩ ∈ |Ω⟩ × |Ω⟩}

|ω⟩ ∼|R⟩|ω′⟩ ⇐⇒R||ω⟩=R||ω′_⟩; (17)

where the following bra-ket-denotations for pairs of bra-points and ket-points are used correspondingly:

⟨ω∗_{, ω}∗′_{|= (⟨ω}∗_|,⟨ω∗′_{|)∈ ⟨Ω| × ⟨Ω|=⟨Ω,Ω|,}

|ω, ω′⟩= (|ω⟩,|ω′⟩)∈ |Ω⟩ × |Ω⟩=|Ω,Ω⟩. (18)

Definition 4(⟨R_|-equivalent classes). Let ⟨R_{| ⊆ ⟨Ω,Ω| be the bra-relation of equivalence on ⟨Ω|.}

Then for any bra-point⟨ω∗| ∈ ⟨Ω|the⟨R_{|-equivalent classof bra-points that⟨}R_{|-equivalent to⟨ω}∗_|:

[⟨ω∗|]⟨R|= {

⟨ω∗′|: ⟨ω∗′| ∼⟨R|⟨ω∗| }

={

is defined.

Definition 5(|R_⟩-equivalent classes). Let |R_{⟩ ⊆ |Ω,Ω⟩ be the ket-relation of equivalence on |Ω⟩.}

Then for any ket-point|ω⟩ ∈ |Ω⟩the|R_{⟩-equivalent classket-points that|}R_{⟩-equivalent to|ω⟩}

[|ω⟩]|R⟩= {

|ω′⟩: |ω′⟩ ∼|R⟩|ω⟩}

={

|ω′⟩:R_|ω′_⟩=R_|ω⟩}⊆ |Ω⟩ (20)

is defined.

Definition 6(bra-quotient-set). The bra-quotient-set of the bra-set ⟨Ω| by equivalence relation ⟨R_|

is the set⟨Ω|/⟨R_{|composed from⟨}R_{|-equivalent classes:}

⟨Ω|/⟨R_|:={_[⟨ω∗_|]⟨R|: ⟨ω∗| ∈ ⟨Ω| }

. (21)

Definition 7(ket-quotient-set). The ket-quotient-set of the ket-set |Ω⟩ by equivalence relation |R_⟩

is the set|Ω⟩/|R_{⟩composed from|}R_{⟩-equivalent classes:}

|Ω⟩/|R_⟩:={

[|ω⟩]|R⟩: |ω⟩ ∈ |Ω⟩ }

. (22)

It is worth emphasizing that a bra-quotient-set and a ket-quotient-set are sets of sets. Each of their elements is itself a set, i.e.⟨Ω|/⟨R_{| ⊆} P_{(⟨Ω|)and|Ω⟩/|}R_{⟩ ⊆} P_{(|Ω⟩), where}P_(⟨Ω|)andP_{(|Ω⟩)is the set of all}

subsets of the set⟨Ω|and of the set|Ω⟩correspondingly.

Definition 8(bra-ket-quotient-set).Thebra-ket-quotient-set of Cartesian product⟨Ω|Ω⟩by equivalence

relation R _{⊆ ⟨Ω|Ω⟩ is the partition of Cartesian product ⟨Ω|Ω⟩ which defined as Cartesian product of}

quotient-set by Minkowski (Cartesian M-product):

⟨Ω|Ω⟩/R_=⟨Ω|/⟨R_{|(×)|Ω⟩/|}R_⟩={

[⟨ω∗|]⟨R|×[|ω⟩]|R⟩: ⟨ω∗|ω⟩ ∈ ⟨Ω|Ω⟩ }

⊆P_{(⟨Ω|Ω⟩), (23)}

composed from Cartesian products of⟨R_{|-equivalent and|}R_{⟩-equivalent classes correspondingly.}

|ω⟩ ∈ |ter(X//XR)⟩

|x′ ⟩=R

⟨ω∗′ |

z }| {

| {z }

|x⟩=R

⟨ω∗ |

⟨ω∗′_|ω⟩

⟨ω∗′_{| ∈ ⟨x}′_|

x′∈/X

R

_{⊆ ⟨}

_Ω

_|

_Ω

_⟩

⟨ω∗_|ω⟩

⟨ω∗_{| ∈ ⟨x|}

x∈X

⟨TerX//X_R|=R||ω⟩

                                                                                                                                                                   

⟨

_Ω

|

⟨

Ω

|

Ω

⟩

| {z }

[image:6.612.113.488.427.734.2]

|

Ω

⟩

Figure 1: Venn diagram of the binary relationR⊆ ⟨Ω|Ω⟩on Cartesian product⟨Ω|Ω⟩with theR-labelling⟨XR|S

XR_{⟩and of three cross-sections of the}

binary relation:R||ω⟩,R|⟨ω∗ |,R|⟨ω∗ ′ |by the ket-point|ω⟩ ∈ |Ω⟩and the bra-points⟨ω∗|,⟨ω∗′| ∈ ⟨Ω|, where the following membership relations:

|X1⟩ |X2⟩ |X3⟩ |X4⟩ |X5⟩

X

c(c) 1

⟩ X

c(c) 2

⟩ X

c(c) 3

⟩ X

c(c) 4

⟩ X

c(c) 5

⟩ ⟨x1|

⟨x2|

⟨x3|

⟨x4|

⟨x5|

⟨x6|

⟨x7|

⟨ x1 | X 1 ⟩ ⟨ x1 | X 2 ⟩ ⟨ x

c|1

X c ( c ) 3 ⟩ ⟨ x

c|1

X c ( c ) 4 ⟩ ⟨ x

c|1

X c ( c ) 5 ⟩ ⟨ x

c|2

X c ( c ) 1 ⟩ ⟨ x2 | X2 ⟩ ⟨ x2 | X3 ⟩ ⟨ x

c|2

X c ( c ) 4 ⟩ ⟨ x

c|2

X c ( c ) 5 ⟩ ⟨ x

c|3

X c ( c ) 1 ⟩ ⟨ x

c|3

X c ( c ) 2 ⟩ ⟨ x3 | X 3 ⟩ ⟨ x3 | X 4 ⟩ ⟨ x

c|3

X c ( c ) 5 ⟩ ⟨ x

c|4

X c ( c ) 1 ⟩ ⟨ x

c|4

X c ( c ) 2 ⟩ ⟨ x

c|4

X c ( c ) 3 ⟩ ⟨ x4 | X4 ⟩ ⟨ x4 | X5 ⟩ ⟨ x5 | X 1 ⟩ ⟨ x5 | X 2 ⟩ ⟨ x5 | X 3 ⟩ ⟨ x

c|5

X c ( c ) 4 ⟩ ⟨ x

c|5

X c ( c ) 5 ⟩ ⟨ x

c|6

X c ( c ) 1 ⟩ ⟨ x6 | X2 ⟩ ⟨ x6 | X3 ⟩ ⟨ x6 | X4 ⟩ ⟨ x

c|6

X c ( c ) 5 ⟩ ⟨ x

c|7

X c ( c ) 1 ⟩ ⟨ x

c|7

X c ( c ) 2 ⟩ ⟨ x7 | X 3 ⟩ ⟨ x7 | X 4 ⟩ ⟨ x7 | X 5 ⟩

⟨xc

1|

⟨xc

2|

⟨xc3|

⟨xc

4|

⟨xc

5|

⟨xc

6|

⟨xc

7|

|X1⟩ |X2⟩ |X3⟩ |X4⟩ |X5⟩

X

c(c) 1

⟩ X

c(c) 2

⟩ X

c(c) 3

⟩ X

c(c) 4

⟩ X

c(c) 5

⟩ ⟨x1|

⟨x2|

⟨x3|

⟨x4|

⟨x5|

⟨x6|

⟨x7|

⟨x1|X1⟩ ⟨x1|X2⟩

⟨

xc

1|X

c(c) 3

⟩ ⟨

xc

1|X

c(c) 4

⟩ ⟨

xc

1|X

c(c) 5

⟩

⟨

xc

2|X

c(c) 1

⟩

⟨x2|X2⟩ ⟨x2|X3⟩

⟨

xc

2|X

c(c) 4

⟩ ⟨

xc

2|X

c(c) 5

⟩

⟨

xc

3|X

c(c) 1

⟩ ⟨

xc

3|X

c(c) 2

⟩

⟨x3|X3⟩ ⟨x3|X4⟩

⟨

xc

3|X

c(c) 5

⟩

⟨

xc

4|X

c(c) 1

⟩ ⟨

xc

4|X

c(c) 2

⟩ ⟨

xc

4|X

c(c) 3

⟩

⟨x4|X4⟩ ⟨x4|X5⟩

⟨x5|X1⟩ ⟨x5|X2⟩ ⟨x5|X3⟩

⟨

xc

5|X

c(c) 4

⟩ ⟨

xc

5|X

c(c) 5

⟩

⟨

xc6|X

c(c) 1

⟩

⟨x6|X2⟩ ⟨x6|X3⟩ ⟨x6|X4⟩

⟨

xc6|X

c(c) 5

⟩

⟨

xc

7|X

c(c) 1

⟩ ⟨

xc

7|X

c(c) 2

⟩

⟨x7|X3⟩ ⟨x7|X4⟩ ⟨x7|X5⟩

⟨xc

1|

⟨xc

2|

⟨xc3|

⟨xc

4|

⟨xc

5|

⟨xc

6|

⟨xc

7|

Figure 2: Bra-ket labelling the Cartesian product⟨Ω|Ω⟩by the binary relationsR( red ) and the complementary binary relationsRc( aqua ); Venn diagram 7x5; terraced bra-events ( red ) and complementary terraced bra-events ( aqua )—left, ket-events ( red ) and complementary ket-events ( aqua )—right. Due to lack of table-space for terraced ket-events forced abbreviations:|X⟩=|ter(X//X_)⟩and|Xc(c)_⟩=|ter(Xc(c)_//X(c)_{)⟩are used}

here.

3 Element-set labelling by a binary relation

Element-set labelling of the Cartesian product⟨Ω|Ω⟩ which will be discussed, is based on the curious fundamental property of measurable binary relationsR_{⊆ ⟨Ω|Ω⟩, which, it turns out, allows each of them}

to define its ownelement-setR_{-labellingof Cartesian product⟨Ω|Ω⟩.}

To see this, we will build two labellings of Cartesian product⟨Ω|Ω⟩by the binary relationR_{⊆ ⟨Ω|Ω⟩and}

by its complementRc _{⊆ ⟨Ω|Ω⟩. Both mutually complementary labellings are the labellings of the same}

Cartesian product⟨Ω|Ω⟩with the help of elements and subsets of some labelling set and some set of its labelling subsets, which justifies their name: element-set labellings. The labelling set, and the set of its labelling subsets both are completely defined by these binary relations, forming theR_{-stock of labelsΛ}R that consists fromR_-labelsλR ∈ΛR; and theRc-stock of labelsΛR, that consists fromRc-labelsλRc ∈ΛRc.

So, we consider:

⋆ theelement-setR_{-labellingby binary relation}R_{⊆ ⟨Ω|Ω⟩of Cartesian product⟨Ω|Ω⟩and three}

quotient-sets⟨Ω|/R_,|Ω⟩/R_{and⟨Ω|Ω⟩/}R_{by this relation with the help of elements and subsets of the labelling set}

XRand the set S XR

⊆P₍XR)of its labelling subsetsX ⊆XR, where both setsXRand S XR

are defined by relationR_{(see definitions below), and}

⋆ theelement-setRc_{-labellingby complementary binary relation}Rc _{=⟨Ω|Ω⟩ −}R_{of the same Cartesian}

product⟨Ω|Ω⟩and three quotient-sets ⟨Ω|/Rc_,|Ω⟩/Rc _{and ⟨Ω|Ω⟩/}Rc _{by this relation with the help of}

elements and subsets of the labelling setX(c_R) ₌ {xc_{: x∈XR} and the set S} X(c_R)

= {Xc(c)_{:X ∈ S} XR} ⊆

P(X(c_R))_{of its labelling subsetsX}c(c)_⊆X(_{R, where both sets}c) X(c_R)_{and S} X (c)

Rare defined by relation_Rc(see definitions below).

R _{⊆ ⟨Ω|Ω⟩, because element-set}Rc_{-labelling serves as its routine «complementary reflection», which is}

easy to construct by looking at theR_{-labelling (see Figire 2).}

3.1 Element-set labelling

The element-setR_{-labelling of Cartesian product⟨Ω|Ω⟩, which uses basic element-set labels (78) from the}

measurable space(Ω,A), begins with the construction of an element-set labelling of the measurable ket space|Ω,A⟩= (|Ω⟩,|A⟩)and its parts with the help of isomorphic images of parts of the measurable space

(Ω,A)3_{. After defining a number of initial concepts within the ket space, it becomes possible to define the} R_{-labelling set}XR⊆ Aand the set of labelling subsets S

XR

⊆P₍XR), in order to complete the construction of the element-set labelling of the ket-space |Ω,A⟩, the bra-space ⟨Ω,A|, and,finally, the bra-ket-space ⟨Ω,A|Ω,A⟩.

3.1.1 Element-set labelling of the ket-space

Definition 9(ket-points). The ket-points |ω⟩ ∈ |Ω⟩ are labelled by point labels ω ∈ Ω.

Definition 10(ket-set). Theket-set |Ω⟩is labelled by the label Ωand defined as a set of all labelled

ket-points:

|Ω⟩={|ω⟩:ω∈Ω}. (24)

Definition 11(ket-subsets). The ket-subsets |x⟩ ⊆ |Ω⟩ are labelled by labels x ⊆ Ω and defined as

subsets, copmosed from corresponding labelled ket-points:

|x⟩={|ω⟩:ω∈x} ⊆ |Ω⟩. (25)

Definition 12(ket-sigma-algebra).Theket-sigma-algebra|A⟩is labelled by the labelAand defined as a

set of measurable ket-subsets|x⟩ ⊆ |Ω⟩labelled by measurable labelsx∈ A:

|A⟩={|x⟩:x∈ A}. (26)

Definition 13(measurable ket-space).Themeasurable ket-space|Ω,A⟩is considered to be a labelled by

the label of measurable space(Ω,A)and defined as the pair|Ω,A⟩= (|Ω⟩,|A⟩).

3.1.2 Two basic labelling sets

Definition 14(basic R-labelling set XR). The basic R-labelling set XR ⊆ A of measurable subset

ofΩis defined by the binary relationR_{⊆ ⟨Ω|Ω⟩as the set of labels}

XR= {

x∈ A: |x⟩=R_|⟨ω∗_|,⟨ω∗| ∈ ⟨Ω|}⊆ A, (27)

ccomposed from measurable subsetsx ⊆ Ωlabelling кет_{-subsets|x⟩ ⊆ |Ω⟩that serve by values of the}

cross-sections:|x⟩=R_|⟨ω∗_|⊆ |Ω⟩of binary relationRby bra-points⟨ω∗| ∈ ⟨Ω|.

Definition 15(basic set S XR _{of R-labelling subsets).} The basic set S XR ⊆ P(_X

R) of R-labelling

subsets of measurable subsets ofΩis defined by the binary relationR_{⊆ ⟨Ω|Ω⟩as the set of set-labels}

S

XR

={

X ⊆XR:∅Ω∈/X,ter(X//XR)̸=∅Ω }

⊆P₍XR), (28)

composed only from labelling subsetsX⊆XRthat do not contain theΩ-empty label:∅Ω∈/X, and number theΩ-nonempty terraced labels:ter(X//XR)̸=∅Ω.

3_{I will not emphasize here that the basis of this isomorphism is theMinkowski principle (M-principle)of constructing an operation}

on sets by means of an isomorphism of operations on elements of these sets, as, for example, in the definition of the operation of

3.1.3 Terraced ket-subsets and its properties

Definition 16(terraced ket-subsets). Theterraced ket-subsets |ter(X//X_{R)⟩ ⊆ |Ω⟩labelled by terraced}

labels

ter(X//XR) = ∩

x∈X

x ∩

x∈XR−X

xc_{⊆Ω, (29)}

aree defined in|Ω⟩forX⊆XRby isomorphic formulas |ter(X//X_R)⟩={

|ω⟩ ∈ |Ω⟩:ω∈ter(X//X_R)}

= ∩

x∈X

|x⟩ ∩

x∈XR−X

|x⟩c⊆ |Ω⟩, (30)

wherexc_{= Ω−xand|x⟩}c

=|Ω⟩ − |x⟩is the set theoretic complements tillΩand till|Ω⟩correspondingly.

Property 2(terrcaed formulas of ket-inversing). The terraced ket-subsets |ter(X//XR)⟩ ⊆ |Ω⟩ are

linked in|Ω⟩with the ket-subsets|x⟩ ⊆ |Ω⟩forx∈XRandX ⊆XRby terraced formulas of ket-inversing (see

[3]):

|ter(X//XR)⟩= ∩

x∈X

|x⟩ ∩

x∈XR−X

|x⟩c⊆ |Ω⟩,

|x⟩= ∑

x∈X

|ter(X//X_{R)⟩ ⊆ |Ω⟩,}

(31)

where|x⟩c=|Ω⟩ − |x⟩is the set theoretic complement till|Ω⟩.

Proof.Thefirst formula of ket-inversing in (31) follows from Definition 16. The second formula follows

from thefirst one since

∑

x∈X

|ter(X//XR)⟩= ∑

x∈X

( ∩

z∈X

|z⟩ ∩

z∈XR−X |z⟩c

)

=|x⟩ ∩ 

 ∑

X⊆XR−{x} 

 ∩

z∈X

|z⟩ ∩

z∈(XR−{x})−X |z⟩c



 



=|x⟩ ∩ 

 ∑

X⊆XR−{x}

|ter(X//XR− {x})⟩ 



=|x⟩ ∩ |Ω⟩=|x⟩.

(32)

Property 3(partition of ket-subset and all ket-set). The terraced ket-subsets |ter(X//XR)⟩ ⊆ |Ω⟩

form a partition of ket-subsets|x⟩ ⊆ |Ω⟩for eachx∈XRby formulas:

|x⟩= ∑

x∈X∈ SXR

|ter(X//XR)⟩ ⊆ |Ω⟩, ₍₃₃₎

in particular,

|Ω⟩= ∑

X∈ SX_R

|ter(X//XR)⟩, ₍₃₄₎

terraced ket-subsets|ter(X//XR)⟩forX ∈ S XR

form a partition of all the ket-set|Ω⟩.

Proofis entirely relied on the isomorphism between the measurable ket-space|Ω,A⟩and measurable

space of labels(Ω,A), in which partitions:

x= ∑

x∈X⊆X

ter(X//X_{) ⊆ Ω,}

Ω = ∑

X⊆X

ter(X//X_),

(35)

3.1.4 Dual element-set labelling bra-space

The element-set labelling of measurable bra-space⟨Ω,A|= (⟨Ω|,⟨A|)generated by the measurable binary relationR _{⊆ ⟨Ω|Ω⟩, is also constructed using basic element-set labels (78) from the measurable space} (Ω,A). However, unlike the isomorphic labelling of the ket-space|Ω,A⟩, this labelling is not isomorphic to

(Ω,A), but is its mapping, which is appropriate to callR_{-dual isomorphism, and talk aboutdual} element-set labelling of a bra-space.

Definition 17(bra-points).Thebra-points⟨ω∗_{|are labelled by point labelsω}∗_∈Ω.

Definition 18(bra-set). The bra-set ⟨Ω| is labelled by the label Ω and defined as the set of all

labelled bra-points:

⟨Ω|={⟨ω∗_|:ω∗_{∈Ω}. (36)}

Definition 19(bra-subset). The bra-subsets ⟨x| ⊆ ⟨Ω| are labelled by labels x ⊆ Ω and defined as

the subsets composed from labelled bra-points⟨ω∗_{| ∈ ⟨Ω|the cross-sectionsR|}

⟨ω∗_|by which coincide with

the ket-subsets|x⟩ ⊆ |Ω⟩lebelled by the same labelx⊆Ω:

⟨x|={⟨ω∗|:R_|⟨ω∗_|=|x⟩} ⊆ ⟨Ω|. (37)

Property 4(bra-subsets as ⟨R_{|-equivalent classes).} Each bra-subset ⟨_x| ∈ ⟨A|_{, x} ∈ XR, is the ⟨R|

-equivalent class[⟨ω∗_|]

⟨R|∈ ⟨Ω|/⟨R|for some⟨ω∗| ∈ ⟨Ω|and each⟨R|-equivalent class[⟨ω∗|]⟨R|∈ ⟨Ω|/⟨R|is the

bra-subset⟨x|for somex∈XR. In other words, the following two assertions are equivalent:

⟨x|= [⟨ω∗|]⟨R| ⇐⇒ ⟨ω∗| ∈ ⟨x|. (38)

Proof.By definitions 18 and 4 the left equality in (38)

{⟨ω∗|:R_⟨ω∗_|=|x⟩}=⟨x|= [⟨ω∗|]_⟨R|={⟨ω∗′|:R|⟨_ω∗′_|=R|⟨_ω∗|} (39)

means thatR_⟨ω∗_|=|x⟩, i.e.⟨ω∗| ∈ ⟨x|. Conversely, if the membership relation on the right-hand side of (38)

holds, then by the same definitions

⟨x|={⟨ω∗|:R_⟨ω∗_|=|x⟩}={⟨ω∗′|:R|⟨_ω∗′_|=R|⟨_ω∗|}= [⟨ω∗|]_⟨R|. (40)

sinceR_|⟨ω∗′_|=R|⟨_ω∗_|=|x⟩for⟨ω∗′|,⟨ω∗| ∈ ⟨x|.

Property 5(partition of a bra-set by bra-subsets). From Property 4 it immediately follows a partition

of the bra-set⟨Ω|by bra-subsets:

⟨Ω|= ∑

x∈XR

⟨x| (41)

since the⟨R_{|-equivalent classes form a partition of the bra-set⟨Ω|.}

Definition 20(terraced bra-subsets). The terraced bra-subsets ⟨TerX//XR| ⊆ ⟨Ω| are labelled by

terraced labels

TerX//XR = ∪

x∈X

x⊆Ω (42)

and defined for each set-labelX ⊆X_{Ras the isomorphic terraced operations}

⟨TerX//XR|= ∪

x∈X

⟨x| ⊆ ⟨Ω| (43)

over the bra-subsets⟨x| ⊆ ⟨Ω|.

as the minimal sigma-algebra which cantains the sets of measurable bra-subsets⟨x| ⊆ ⟨Ω|labelled by measurable labelsx∈ A:

⟨A|=σ(

{⟨x|:x∈ A})

. (44)

Definition 22(measurable bra-space). The measurable bra-space ⟨Ω,A| is labelled by the label of

measurable space(Ω,A)and defined as the pair⟨Ω,A|= (⟨Ω|,⟨A|).

Property 6(partition of terraced bra-subset and all bra-set). The bra-subsets ⟨x| ⊆ ⟨Ω| form a

partition of terraced bra-subset⟨TerX//XR| ⊆ ⟨Ω|for eachX ⊆XRby formulas: ⟨TerX//XR|=

∑

x∈X

⟨x| ⊆ ⟨Ω|, ₍₄₅₎

in particular,

⟨TerXR//XR|= ∑

x∈XR

⟨x| = ⟨Ω|, ₍₄₆₎

the bra-subsets⟨x|forx∈XRform a partition of all the bra-set⟨Ω|(see Property 5).

Proof.Since the bra-subsets ⟨x| ⊆ ⟨Ω| are defined (38) by classes of equivalent cross-sections of the

binary relationR_{they do not intersect:⟨x| ∩ ⟨y|=}∅⟨Ω|forx̸=yand (45) follows from (43).

Property 7(terraced formulas of bra-inversing). The terraced bra-subsets⟨TerX//XR| ⊆ ⟨Ω|are linked in

⟨Ω|with the bra-subsets⟨x| ⊆ ⟨Ω|forx∈XRandX ⊆XRterraced formulas of bra-inversing (see [3]):

⟨TerX//XR|= ∑

x∈X

⟨x| ⊆ ⟨Ω|,

⟨x|= ∩

x∈X

⟨TerX//XR| ∩

x∈XR−X

(⟨Ω| − ⟨TerX//XR|)⊆ ⟨Ω|.

(47)

Proof. Thefirst formula of bra-inversing in (47) follows from Property 6, the formula (45). The second

one follows from thefirst one By virtue of Lemma 1 since the bra-subsets⟨x| ⊆ ⟨Ω| forx ∈ XR satisfy conditions of this lemma forming a partition of⟨Ω|(see the formula (46)).

Lemma 1(dual formulas of terraced inversing).Let X_{⊆ Abe a set of measurable subsets ofΩforming a}

partition ofΩ:

Ω =∑

x∈X

x, (48)

and theX-partial sums of these subsets have the notation forX ⊆X_:

TerX//XR = ∑

x∈X

x⊆Ω. (49)

Then forx∈X_{the dual formulas of terraced inversing are valid:}

x= ∩

x∈X

TerX//X ∩

x∈X−X

(

TerX//X )c

⊆Ω. (50)

Proof.First we note that

(

TerX//X )c

= Ω−TerX//XR= ∑

x∈Xc

x=TerXc_//XR, (51)

whereXc_{=X−Xis the set theoretic complement tillX. Going to complements tillΩin both parts of the}

equation (50) we get the equivalent formula:

xc= ∪

x∈X

TerXc_//X ∪

x∈Xc

in which all partial sumsTerXc_//XRandTerX//XR, included in the union on the right, do not containx∈X. In fact, the sumTerXc_//XR=

∑

z∈Xc

zforx∈Xdoes not contain the termx. And the sumTerX//XR= ∑

z∈X

zfor

x∈Xc_{also does not contain the termx. Hence we get what is required:}

∪

x∈X

TerXc_//X ∪

x∈Xc

TerX//X= Ω−x=xc. (53)

3.1.5 Terraced ket-subsets as equivalent classes and other properties of the element-set labelling by a binary relation

Property 8.The following assertions are equivalent forx∈XR:

|ω⟩ ∈ |x⟩ ⇐⇒ ⟨x| ⊆R_{||ω⟩, (54)}

|ω⟩∈ |/ x⟩ ⇐⇒ ⟨x| ∩R_||ω⟩=∅⟨Ω|. (55)

Proof. The first equivalence (54). Note that by Definition 19 |ω⟩ ∈ |x⟩ then and only then when

|ω⟩ ∈ R_|⟨ω∗_|for all ⟨ω∗| ∈ ⟨x|. By virtue of duality of cross-sections (see Property 1) |ω⟩ ∈ R|⟨_ω∗_| for all

⟨ω∗_{| ∈ ⟨x| then and only then when⟨ω}∗_{| ∈ R|}

|ω⟩ for all⟨ω∗| ∈ ⟨x|. But this means that⟨x| ⊆ R||ω⟩. The

second equivalence (55). Note that by Definition 19|ω⟩∈ |/ x⟩then and only then when|ω⟩∈/ R_|⟨ω∗_|for all

⟨ω∗_{| ∈ ⟨x|. By virtue of duality of cross-sections (see Property 1)|ω⟩∈/R|}

⟨ω∗_|for all⟨ω∗| ∈ ⟨x|then and only

then when⟨ω∗_{|∈/ R|}

|ω⟩for all⟨ω∗| ∈ ⟨x|. But this means that⟨x| ∩R||ω⟩=∅⟨Ω|.

Property 9.The following assertions are equivalent forx∈XandX ∈ S XR

:

⟨ω∗| ∈ ⟨x| ⇐⇒ R_|⟨ω∗_|=|x⟩, (56)

|ω⟩ ∈ |ter(X//XR)⟩ ⇐⇒ R||ω⟩=⟨TerX//XR|. (57)

Proof.Thefirst equivalence (56) is valid by Definition 19. The second equivalence (57). Note that by

Definition 16|ω⟩ ∈ |ter(X//XR)⟩then and only then when|ω⟩ ∈ |x⟩for allx∈X and|ω⟩∈ |/ x⟩for allx /∈X. By Property 8 this means that⟨x| ⊆ R_{||ω⟩ for allx ∈ X and⟨x| ∩}R_{||ω⟩ =} ∅⟨Ω| for allx /∈ X. But this is

equivalent to the equalityR_{||ω⟩ =} ∑

x∈X

⟨x|due to the fact that the bra-subsets⟨x| ⊆ ⟨Ω|as⟨R_|-equivalent

classes pairwise disjoint in⟨Ω|. Applying Definition 20 we obtain the required equality:R_{||ω⟩=⟨TerX//X}

R|.

Property 10(terraced ket-subsets as |R_{⟩-equivalent classes).} Each terraced ket-subset |ter_(X//XR)⟩ ∈

|A⟩, X ∈ S XR

is the|R_{⟩-equivalent class[|ω⟩]|R⟩ ∈ |Ω⟩/|}R_{⟩for some|ω⟩ ∈ |Ω⟩and each |}R_{⟩-equivalent class} [|ω⟩]|R⟩∈ |Ω⟩/|R⟩is the terraced ket-subset|ter(X//XR)⟩for someX ∈ S

XR

. In other words, the following two assertions are equivalent:

|ter(X//XR)⟩= [|ω⟩]|R⟩ ⇐⇒ |ω⟩ ∈ |ter(X//XR)⟩. (58)

Proof.The second equivalence (57) in Property 9 defines a terraced ket-subset as a subset of ket-points:

|ter(X//XR)⟩= {

|ω⟩:R_{||ω⟩=⟨TerX//}XR| }

⊆ |Ω⟩, (59)

which coincides with the|R_{⟩-equivalent class[|ω⟩]|R⟩for any ket-point|ω⟩ ∈ |ter(X//}XR)⟩since

{

|ω⟩:R_{||ω⟩=⟨TerX//}XR| }

={

|ω′⟩: R_||ω′_⟩=R||_ω⟩=⟨Ter_X//XR| }

= [|ω⟩]|R⟩. (60)

Property 11(three partitions of binary relation in the own element-set labelling). For the binary

columns»X ∈ S XR

and «by elements», which in its element-setR_{-labelling have the form:}

R ₌ ∑

x∈X_R

x̸=∅Ω

⟨x|x⟩ (61)

—partition «by rows»x,

R ₌ ∑

X∈ SX_R

⟨TerX//XR|ter(X//XR)⟩ (62)

—partition «by columns»X,

R ₌ ∑

X∈ SXR ∑

x∈X∈ SXR

⟨x|ter(X//XR)⟩ (63)

—partition «by elements»: «by columns»Xand «by rows»x,

R ₌ ∑

x∈XR ∑

x∈X∈ SX_R

⟨x|ter(X//XR)⟩ (64)

—partition «elements»: «by rows»xand «by columns»X.

Property 12(formulas of dual link between bra-subsets and ket-subsets). For each X ∈ S XR

the terraced bra-subset ⟨TerX//XR| ⊆ ⟨Ω| serves as general value of all equivalent cross-sections R||ω⟩ by

ket-points|ω⟩ ∈ |ter(X//XR)⟩from the terraced ket-subset|ter(X//XR)⟩,i.e. forX ∈ S XR

the formula of dual link is valid:

|ter(X//XR)⟩={|ω⟩:R||ω⟩=⟨TerX//XR|} ⊆ |Ω⟩. (65)

For eachx∈ X_{R the ket-subset|x⟩ ⊆ |Ω⟩serves as general value of all equivalent cross-sections}R_|⟨ω∗_| by bra-points⟨ω∗_{| ∈ ⟨x|from the bra-subset⟨x| ⊆ ⟨Ω|, i.e., forx∈X}

Rthe formula od dual link is valid:

⟨x|={⟨ω∗_|:R|⟨

ω∗_|=|x⟩} ⊆ ⟨Ω|. (66)

Proof. Thefirst formula (65) follows from (57) in Property 9 (see also (59)). The second formula (66) is

correct by Definition 19.

Property 13(labelling the cross-sections of binary relation).

⟨ω∗| ∈ ⟨x| ⇐⇒ R_|⟨ω∗_|=

∑

x∈X

|ter(X//XR)⟩, (67)

|ω⟩ ∈ |ter(X//XR)⟩ ⇐⇒ R||ω⟩= ∑

x∈X

⟨x|. (68)

Proof. A labelling the cross-sections ofR_{by bra-points⟨ω}∗_{| ∈ ⟨x|is thefirst formula (56) in Property 9}

taking into account the partition of a ket-subset by terraced ket-subsets (33). A labelling the cross-sections ofR_{by ket-points|ω⟩ ∈ |ter(X//}XR)⟩is the second formula (57) in Property 9.

Definition 23(element-set labelling a quotient-set by binary relation).They say that

⟨Ω|/R_=⟨X_{R|={⟨x|:x∈}X_{R}, (69)}

a labelling the bra-quotient-set⟨Ω|/R_{by binary relation}R_{⊆ ⟨Ω|Ω⟩, in which labelsx∈}XRof the labelling setXRlabel allbra-subsets⟨x| ∈ ⟨Ω|/Rof this quotient-set;

|Ω⟩/R₌ S

XR⟩

={|ter(X//XR)⟩:X ∈ S XR}

, (70)

a labelling the ket-quotient-set|Ω⟩/R_{by binary relation}R_{⊆ ⟨Ω|Ω⟩, in which set-labelsX ∈ S} XR

from the set of labelling sets S XR

label theterraced ket-subsets|ter(X//XR)⟩ ∈ |Ω⟩/Rof this quotient-set;

⟨Ω|Ω⟩/R₌⟨XR S

XR⟩

={⟨x|ter(X//XR)⟩:x∈XR, X ∈ S XR}

a labelling the bra-ket-quotient-set ⟨Ω|Ω⟩/R_{by binary relation} R _{⊆ ⟨Ω|Ω⟩, in which pairs(x, X), where}

x ∈ XR is an element of the labelling setXR, andX ∈ S XR

is the element from the set S XR

of labelling subsets, label all thebra-ket-subsets⟨x|ter(X//XR)⟩ ∈ ⟨Ω|Ω⟩/Rof this quotient-set.

Predefinition 2(R-labelling by basic element-set labels). R_{-labelled by basic element-set labels}

λ∈ Λparts of bra-space and ket-space are supplied with general notation⟨λR|and|λR⟩, a list of which can be found in Appendix on page 122.

3.1.6 Measurable binary relation as a membership relation

Theorem 1(measurable binary relation as a membership relation). Any measurable binary relation

R_{⊆ ⟨Ω|Ω⟩on Cartesian product⟨Ω|Ω⟩is equivalent to the membership relation}

R

⟨XR| SXR⟩= {

⟨x|ter(X//XR)⟩:x∈X }

⊆⟨XR S

XR⟩

(72)

in the element-setR_-labelling⟨XR S

XR⟩

of the quotient-set⟨Ω|Ω⟩/R_{. In other words,}

R₌{_⟨ω∗_{|ω⟩ ∈ ⟨Ω|Ω⟩: ⟨ω}∗_{|ω⟩ ∈ ⟨x|ter(X//}XR)⟩ ∈R_⟨XR|SXR_⟩ }

⊆ ⟨Ω|Ω⟩. (73)

Proofis based on the equivalence

⟨x|ter(X//XR)⟩ ⊆R ⇐⇒ x∈X, (74)

from which it follows that the membership relation⟨ω∗_{|ω⟩ ∈ R is equivalent to the fulfillment of two} membership relations:⟨ω∗_{|ω⟩ ∈ ⟨x|ter(X//X}

R)⟩andx∈Xwhich proves the theorem.

4 Appendix

List 1(basic element-set labels). The basic element-set labels λ ∈ Λ, or simply basic labels, are

terraced set theoretic operations4_{over them, are equipped with their own names}5_: λ=                                                                               

Ω label of a set,

A label of a sigma-algebra,

(Ω,A) label of a measurable space,

X⊆ A set-label,

S

X

⊆P₍X_{) set of set-labels,}

ω, ω∗_{∈Ω, point label,}

x∈X_{, x⊆Ω label,}

xc _∈X(c)_{, x}c_{⊆Ω label,}

∅Ω⊆Ω Ω-empty label,

X ⊆X _set-label,

Xc(c)_⊆X(c) _set-label,

∅X

⊆X _{Ω-empty set-label,}

ter(X//X_{) =} ∩

x∈X

x ∩

x∈X−X

xc⊆Ω terraced label,

ter(Xc(c)//X(c))_=ter(X//X_{)⊆Ω terraced label,}

TerX//X= ∪

x∈X

x⊆Ω terraced label,

TerXc(c)_//X(c)=

∪

x∈X−X

xc⊆Ω terraced label.

(75)

taking into account the definition of terraced sets and the validity of terraced equalities inΩ(see [3]):

ter(Xc(c)_//X(c))₌ ∩

xc_∈Xc(c)

xc ∩

xc_∈X(c)_−Xc(c)

x= ∩

x∈X

x ∩

x∈X_−X

xc _{=ter(X//X)⊆Ω,}

TerXc(c)_//X(c) =

∪

xc_∈Xc(c)

xc= ∪

x∈X−X

xc⊆Ω.

(76)

List 2(R-labelling by basic element-set labels). R_{-labelled by basic element-set labels λ ∈ Λ parts}

of the bra-space and the ket-space, is equipped by the following denotations:

⟨λR|=

                                        

⟨Ω| bra-set,

⟨A| bra-sigma-algebra,

⟨Ω,A| measurable bra-space,

⟨XR| bra-quotient-set,

⟨ω∗_{| ∈ ⟨Ω| bra-point,} ⟨x| ⊆ ⟨Ω|, x∈XR bra-subset, ⟨xc_{| ⊆ ⟨Ω|, x∈X}

R complementary bra-subset,

⟨TerX//XR|= ∑

x∈X

⟨x| ⊆ ⟨Ω|, X∈ S XR

terraced bra-subset,

⟨Ter_Xc(c)_//X(c_R)|=⟨Ω| − ⟨TerX//XR| ⊆ ⟨Ω|, X∈ S XR

complementary terraced bra-subset;

(77)

4_{In [3, 2007] theterraced set theoretic operationof the 1-st type is defined as the subsetter(X//X) =} ∩

x∈X x ∩

x∈X−X

(Ω−x)⊆Ω,

and theterraced set theoretic operationof the 5-th type as the subsetTerX//X=

∪

x∈X x⊆Ω.

|λR⟩=

                    

                   

|Ω⟩ ket-set,

|A⟩ ket-sigma-algebra,

|Ω,A⟩ measurable ket-space,

| S XR

⟩ ket-quotient-set,

|ω⟩ ∈ |Ω⟩, ket-point,

|x⟩ ⊆ |Ω⟩, x∈XR ket-subset, |xc_{⟩ ⊆ |Ω⟩, x∈X}

R complementary ket-subset,

|ter(X//XR)⟩= ∩

x∈X

|x⟩ ∩

x∈XR−X

|x⟩c⊆Ω, X∈ S XR

terraced ket-subset,

|ter(Xc(c)//X(c_R))_⟩=|ter(X//XR)⟩ ⊆ |Ω⟩, X∈ S XR

complementary terraced ket-subset.

(78)

List 3(R-labelling the bra-ket-set and its parts). Each measurable binary relation R _{⊆ ⟨Ω|Ω⟩}

characterizes aelement-set labellingthe bra-ket-set⟨Ω|Ω⟩and its parts which are defined and called by the following way:

1) thebra-point⟨ω∗_{| ∈ ⟨Ω|and theket-point|ω⟩ ∈ |Ω⟩, labelled by labelsω}∗_{∈Ωandω∈Ω;} 2) thequotient-sets:

⟨Ω|/R_=⟨XR|={⟨x|: x∈XR},

|Ω⟩/R₌

S

XR⟩

={|ter(X//XR)⟩: X∈ S XR}

,

⟨Ω|Ω⟩/R₌⟨XR S

XR⟩

=⟨X_{R| × | S} XR

⟩={⟨x|ter(X//XR)⟩: x∈XR, X ∈ S XR}

;

(79)

thecomplementary quotient-sets:

⟨Ω|/Rc_=⟨X(_R|c) _={⟨xc_|:x∈X_R},

|Ω⟩/Rc₌ S

X(c) R ⟩

={|ter(Xc(c)//X(c)

R)⟩: X∈ S

XR}

,

⟨Ω|Ω⟩/Rc₌⟨X(c)

R S

X(c) R ⟩

=⟨X(c)

R| × | S X(c)

R⟩= {

⟨xc_|ter(Xc(c)_//X(c)

R)⟩:x∈XR, X∈ S

XR}

;

(80)

3) thebra-subset

⟨x|={⟨ω∗|:R_|⟨ω∗_|=|x⟩} (81)

consists from bra-points⟨ω∗_{| ∈ ⟨Ω|, the cross-sectionsRby which is equivalent to the ket-subset|x⟩ ⊆ |Ω⟩} that is labelled by the sameR_-labelsx∈X_R;

4) theterraced ket-subset

|ter(X//XR)⟩={|ω⟩:R||ω⟩=⟨TerX//XR|} (82) consists from ket-points |ω⟩ ∈ |Ω⟩, the cross-sections R _{by which are equivalent to the ket-subset}

⟨TerX//XR| ⊆ ⟨Ω|that labelled by the sameR-labelsX∈ S XR

. 5) thecomplementary bra-subset

⟨xc|={⟨ω∗|:Rc_|⟨ω∗_|=|xc⟩} (83)

consists from bra-points⟨ω∗_{| ∈ ⟨Ω|, the cross-sectionsR}c_{by which are equivalent to the ket-subset|x}c_{⟩ ⊆}

|Ω⟩thatRc_{-labelled by the same labelsx∈XR;}

6) thecomplementary terraced ket-subset

|ter(Xc(c)//X(c_R)₎⟩={|ω⟩:Rc_{||ω⟩=⟨Ter}

Xc(c)_//X(c_R)|} (84) consists from ket-points |ω⟩ ∈ |Ω⟩, the cross-sections Rc _{by which are equivalent to the bra-subset}

⟨Ter_Xc(c)_//X(c) R

|thatRc_{-labelled by the same labelX ∈ S} XR . 7) thebinary quotient-relationR

⟨XR| SXR⟩⊆ ⟨Ω|Ω⟩/Ris the binary relation

R

⟨XR|S

X_R

⟩= {

⟨x|ter(X//X_{)⟩:x∈X;x∈}XR, X∈ S XR}

⊆ ⟨X_{R| S} XR

on the quotient-set⟨XR| S XR

⟩ = ⟨Ω|Ω⟩/R_{, which is equivalent to the membership relation «x ∈ X» on}

XR× S XR

.

8) thecomplementary binary quotient-relationRc

⟨XR|S

X_R

⟩⊆ ⟨Ω|Ω⟩/R

c_{is the binary relation}

Rc

⟨XR|SXR⟩= {

⟨xc_{| |ter(X}c(c)_//X(c)

R)⟩⟩:x

c _∈Xc(c)_;x∈X

R, X∈ S

XR R

}

⊆ ⟨X_{R| S} XR

⟩ (86)

on the quotient-set⟨X_{R| S} XR

⟩=⟨Ω|Ω⟩/Rc_{, which is equivalent to the membership relation «x}c _∈Xc(c)_{» on}

XR× S XR

.

List 4(general denotations of R-labelled subsets).The measurable binary relationR_{⊆ ⟨Ω|Ω⟩generates}

theR_{-labelling of subsets from the bra-ket-set⟨Ω|Ω⟩, the bra-set ⟨Ω|, and the ket-set⟨Ω|. Some of these}

subsets, and also someCartesian productsof these subsets have a common name:R_{-labelled subsets, and}

corresponding common denotations:

⟨λR|=

 



⟨x|, x∈XR,

⟨TerX//XR|, X∈ S XR

,

|λR⟩=

 



|x⟩, x∈XR,

|ter(X//X_{R)⟩, X ∈ S} XR

,

⟨λR|λ′R⟩=⟨x|ter(X//XR)⟩, x∈XR, X∈ S XR

(87)

They say that theR_{-labelledby the same labelλ}R ∈ Λbra-subset⟨λR|and the ket-subset|λR⟩, and also

R_{-labelled by terraced labelsλ}R ∈ Λthat numbered by the same set-label, the terraced bra-subset⟨λR| and the terraced ket-subset|λR⟩arebra-ket-dual to each other and form thepair of bra-ket-dual subsets

in the form of Cartesian product

⟨λR|λR⟩=⟨λR| × |λR⟩=

 



⟨x|x⟩, x∈X_R,

⟨TerX//XR|ter(X//XR)⟩, X ∈ S XR

.

(88)

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2:131–229, 1910.

[3] O. Yu. Vorobyev. Eventology. Siberian Federal University, Krasnoyarsk, Russia, 2007 (in Russian), 435p.,

https://www.academia.edu/179393/.

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Vorobyev ed.):48–99, 2016 (in English, abstract in Russian); ISBN 978-5-9903358-6-8,https://www.academia.edu/34357251.

[5] O. Yu. Vorobyev. Postulating the theory of experience and of chance as a theory of co∼events (co∼beings). In.Proc. of the