[
Abstract
In the theory of conditional sets, many classical theorems from areas such as analysis, probability theory or measure theory are lifted to the conditional framework, often to be applied in areas such as financial mathematics or optimization. The frequent experience that such theorems can be proved by ‘conditionalizations’ of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on secondorder arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, settheoretical topology or uncountable set theory, see e.g. the introduction of [40]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including the BolzanoWeierstraß theorem, the HeineBorel theorem, the Riesz representation theorem and Brouwer’s fixed point theorem.
Transfer principle]A transfer principle for secondorder arithmetic,
and applications
M Carl]Merlin Carl
\givennameMerlin
\surnameCarl
\urladdr
A Jamneshan]Asgar Jamneshan
\givennameAsgar
\surnameJamneshan
\urladdr
\subjectprimarymsc200003C90,03F35
\subjectsecondarymsc200028B20,93E20
\arxivreference
\arxivpassword
In the theory of conditional sets, many classical theorems from areas such as analysis, probability theory or measure theory are lifted to the conditional framework, often to be applied in areas such as financial mathematics or optimization. The frequent experience that such theorems can be proved by ‘conditionalizations’ of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on secondorder arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, settheoretical topology or uncountable set theory, see e.g. the introduction of [40]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including the BolzanoWeierstraß theorem, the HeineBorel theorem, the Riesz representation theorem and Brouwer’s fixed point theorem.
1 Introduction
Fixing a probability space , one can distinguish between probabilistic and deterministic objects such as a deterministic real number which is an element of and a random real number which is a measurable function . Extending such reasoning, one might speak of a ‘random’, ‘stochastic’ or ‘conditional’ version of a ‘deterministic’, ‘classic’ or ‘standard’ theorem which expresses a randomization of its statement. To illustrate, a random or conditional version of the BolzanoWeierstraß theorem states that for every sequence of random real numbers such that almost surely, there exists a strictly increasing sequence of random integers such that if the sequence is ‘randomly’ parametrized by , then it converges almost surely to a unique random real number. Experience showed that many classical theorems have such conditional analogues such as the BolzanoWeierstraß theorem, the HahnBanach extension and separation theorems, the Brouwer fixed point theorem, see e.g. [8, 13, 16, 17, 30]. It is thus tempting to aim for a general transfer principle that allows one to ‘import’ classical theorems into a conditional setting.
In section 3, we prove such a transfer principle based on secondorder logic. Strong arguments have been put forward in favor of the claim that secondorder logic is a satisfying formal framework for the bulk of ‘classical’ mathematics, see e.g. the introduction to [40]; and this was impressively confirmed by the results of reverse mathematics. More precisely, we prove that any consequence of the secondorder axiomatic system of arithmetical comprehension ACA, which has a secondorder comprehension axiom for formulas in which all quantifiers range over natural numbers (see [40]), also holds conditionally. To this end, we show (i) that the axioms of ACA hold in the structure with first order part and secondorder part its conditional power set with truth value and (ii) that truth value is preserved by the usual deduction rules of secondorder predicate calculus. We expect that, with a certain amount of extra technical effort, this can be extended to a considerably stronger transfer principle for full secondorder logic.
In section 4, we verify that the transfer principle yields a conditional version of classical theorems many of which were proved ‘by hand’ previously^{1}^{1}1The latter practice provides some useful insight into the transfer process.. On the other hand, a conditional version of a classical theorem is oftentimes also a theorem about a more involved situation in a classical setting. For instance, a conditional version of the BolzanoWeierstraßtheorem is also a theorem securing existence of certain almost surely converging subsequences of a bounded sequence of realvalued measurable functions. In this classical form under the name of a ‘randomized’, ‘stochastic’ or ‘measurable’ version of the BolzanoWeierstraßtheorem this statement was proved in [19, 33], motivated by applications in mathematical economics.
We will be also interested in the reverse direction. Starting with the real numbers in a ‘conditional’ model of ACA, it can verified that certain classical structures have a well defined ‘conditional’ meaning. For example, the standard space of realvalued Borel measurable functions on finite measure space modulo almost everywhere equality are the real numbers in a ‘conditional’ model of ACA. More generally, if is a complete separable metric space and is the space of valued Borel measurable functions on modulo almost everywhere equality, we show that can be identified with a complete separable metric space in a ‘conditional’ model of ACA. The space reflects a measurable parametrization of the elements of relative to a base space . A measurable parametrization is a constituent part of a conditional model^{2}^{2}2Therefore such models might equally be called “stochastic” or “measurable” models of an axiomatic system.. From an external perspective, one may view the transfer principle as a device which parametrizes classical theorems in a measurable way relative to a fixed measure space. In particular, the application of a conditional version of classical theorems (as obtained from the transfer principle) preserve measurability, and thus providing an alternative to uniformization theorems in descriptive set theory (see e.g. [34, 36, 37]) whenever one restricts attention to almost everywhere Borel selections. For example, we show that the maximum theorem in a randomized model is equivalent to a maximum theorem for normal integrands. In particular, compact subsets of a Euclidean space in a randomized model are uniquely related to compactvalued maps. We will draw a connection to LebesgueBochner spaces and probabilistic analysis as well.
There is a practical interest in such modeltheoretic results as the transfer principle is a highly efficient tool which replaces the tedious work of proving a conditional version classical results which are relevant in applications. Existing areas of application include probability ([30]), mathematical economics ([3, 5, 6, 12, 18, 19, 21, 27, 33]), optimal stochastic control ([7, 9, 31]), random set theory, LebesgueBochner spaces and vector duality ([15]), vector optimization ([23]) and probabilistic analysis ([24, 32]).
Conditional set theory [13], which is used to build a ‘conditional’ model of ACA, is conceptually closely related to Booleanvalued models and topos theory. In fact, a conditional model of ACA is a Booleanvalued model of ACA by changing to the measure algebra associated to an underlying base space . The constructive approach of secondorder arithmetic permits us to explore the semantics in a Booleanvalued model, and through this understanding build a relationship to structures and theorems in a standard model.
In [2], a categorical equivalence between conditional sets and a certain class of Booleanvalued sets is discussed. The correspondence is based on ZFC set theory rather than secondorder arithmetic. We believe that our transfer principle is of independent interest, as it is (1) an explicitly formulated theorem; (2) secondorder arithmetic allows a more direct and convenient modeling of relevant mathematical notions than set theory; for example, natural numbers and real numbers are treated as primitive objects and not as complicated sets; (3) it is not argued in [2] how one can deduce a transfer principle from a categorical equivalence and (4) the correspondence in [2] excludes local subsets which are necessary to prove a transfer principle as shown in the proof of theorem 3.4 below.
2 Preliminaries
In the language of secondorder arithmetic (see [40] for an introduction), we distinguish between number variables, traditionally written in lowercase Latin letters and set variables, usually written as uppercase Latin letters . Moreover, we have two constant symbols and , two binary function symbols and and a binary relation symbol . Between numbers and sets exists an element relation . The firstorder terms are number variables, constant symbols and expressions of the form and for firstoder terms and . The atomic formulas are , and where and are firstorder terms and is a set variable. The remaining formulas are obtained from atomic formulas by the use of propositional connectives and number and set quantifiers.
The axiomatic system of arithmetical comprehension (ACA) consists of the axioms for discretely ordered semirings, together with the secondorder induction principle and an axiomatic scheme saying for any secondorder formula containing only firstorder quantifiers that there is a set of all natural numbers of which holds, see [40]. By the cumulative results of reverse mathematics, ACA is a sufficient axiomatic basis for a great number of theorems from classical mathematics; many examples can be found in [40].
Throughout fix a finite measure space and let denote its ideal of null sets. We always identify whenever where denotes symmetric difference. The resulting quotient Boolean algebra has the following relevant properties:

Completeness: Any family in has a union and an intersection in ;

Countable chain condition: Any pairwise disjoint family in is at most countable;
see e.g. [22, Chapter 31] for a reference. We will always identify two functions and on with the same codomain if . For a function on and , we write for the restriction of to . For any Polish space , let denote the space of Borel functions . For a measurable partition and a countable family in for a Polish space , we write for the unique element with for all . In particular, denotes the space of realvalued functions, and , and denote its subsets consisting of functions with values in , the integers and the rational numbers, respectively. Recall that is a Dedekind complete Riesz lattice where addition, multiplication and order are defined pointwise, see e.g. [20]. By an abuse of language, we also denote by and the functions with constant values and respectively.
We introduce the conditional power set of , see [13] for an introduction to conditional set theory.
Definition 2.1.
A set is said to be stable under concatenations, or stable for short, if it is not empty and for all measurable partitions and countable family in . The conditional power set of is the collection
where . We write .
We define a concatenation of a countable family in and a measurable partition by
where
Let us define conditional intersection and conditional complement. We follow the presentation in [30, Section 2]. Let . The conditional intersection of and is defined as
(2.1) 
where
The conditional complement of is the conditional subset
(2.2) 
where
By applying an exhaustion argument, it can be derived from stability that and are attained, and it can be checked that and are stable sets as well. We conclude that the conditional intersection and conditional complement are well defined, see e.g. [30] for a complete argument.
We introduce a conditional element relation between and its conditional power set.
Definition 2.2.
The conditional element relation is the function
i  
The union of is attained. Indeed, from [22, Section 30, Lemma 1] we know that there exists a countable family in such that . Let and for which defines a measurable partition of . Since for all , the claim follows from stability of . Further, by inspection, we have
(2.3) 
3 A Transfer Principle for ACA
Our aim is to prove that every consequence of ACA holds in the structure with truth value , i.e. that is a ‘conditional model’ of ACA. We start by explaining the evaluation of terms and formulas in .
Let denote the collection of all number variables and let denote the collection of all set variables. Let be a function with domain such that and , and let be a firstorder term. Such a is called a ‘conditional assignment’. Then , the evaluation of , is defined recursively as

for ,

for ,

for ,

for .
For firstorder terms and and a secondorder variable , the conditional evaluation of atomic formulas is defined by

,

,

.
The conditional evaluation of composite and quantified formulas is defined by

,

,

,

.
The remaining composite and quantification formulas are defined in the obvious way.
We have the following maximum principle, also known from Booleanvalued models, see e.g. [4, Chapter 1].
Proposition 3.1.
Let be a conditional assignment and let and be formulas in . Then there exist and such that
Proof.
We may assume that . We find a countable family such that . Form a measurable partition from , still denoted by . Put . Then it holds that
for all which implies . The second claim can be shown analogously by using concatenations in . ∎
We will now adapt the usual notion of the correctness of a sequent to the conditional context.
Definition 3.2.
If and are sets of secondorder formulas, then is called a sequent. The conditional validity of a sequent with respect to a conditional assignment is defined by
and is said to be correct if and only if
for all assignments . An inference rule is a pair consisting of a finite sequence of sequents and a single sequent written as , and it is said to be correct if and only if the correctness of follows from the correctness of .
We want to apply the inference rules for secondorder logic given by Takeuti in [41, p. 910 and p. 135136]. By Boolean arithmetic, one can directly check that all structural and logical rules are correct. We illustrate this for the first weakening rule
(3.1) 
For each assignment , one has
which leads to the correctness of (3.1). By Proposition 3.1, the left universal quantification rule
where is a term, is correct. As for the correctness of the right universal quantification rule
where does not occur freely in , assume
(3.2) 
for all conditional assignments, and let be an arbitrary conditional assignment. We want to show that
(3.3) 
By Proposition 3.1, there exists such that variant of that maps to . Then (3.3) follows from (3.2) since does not occur freely in , so that the first two sets of the union remain unchanged. Analogously, one shows the firstorder existential quantification inference rules. The secondorder quantifier inference rules are only relevant for secondorder variables, as predicate constants do not appear in our language. The inference rules for secondorder quantification can hence be proven analogously to the corresponding firstorder rules. Thus, we obtain: . Choose a
Lemma 3.3.
If is any deduction rule of secondorder sequent calculus and are correct, then is correct. In particular, if all elements of hold in with truth value and is derivable with the rules of sequent calculus, then holds in with truth value .
Theorem 3.4.
All the axioms of ACA attain the value in the structure for all conditional assignments.
Proof.
Let be an arbitrary conditional assignment. The verification of the basic axioms (see [40, p. 4]) is immediate from the definitions. For the sake of completeness, we provide the elementary arguments below.

.

Since , it follows from that .

Similarly, one can verify that , , , , and .

is equivalent to which means that .
As for the secondorder induction scheme, we have to verify that
By conditionally evaluating the previous formula and rewriting it by using Boolean arithmetic, we must verify that , where
But this is immediate from the stability of .
Finally, we verify the arithmetical comprehension scheme, that is, we want to show that
for any arithmetical^{3}^{3}3Recall that a formula of is said to be arithmetical if it contains no set quantifiers, see e.g. [40]. formula in which does not occur freely. By Proposition 3.1, is attained. Suppose for a moment that
satisfies stability. Then
(3.4) 
for all . Indeed, for , let be such that where , and put . By stability of , we have
By Boolean arithmetic, it follows from (3.4) that
which proves comprehension. Thus it remains to verify that is stable under concatenations for all formulas which we will prove by an induction on arithmetical formulas. First, since addition and multiplication commute with concatenations^{4}^{4}4That is, for ., for any firstorder term , by an induction on terms, one has
for all measurable partitions and every countable family in . Since also order and concatenations commute and due to (2.3), is stable for all atomic formulas .
Let and be two arithmetical formulas such that . Then we have
Moreover, for a negation one obtains
Finally, let be arithmetical and . Clearly, . By the established, we can already define a pairing function, product of stable sets and their projections, see e.g. [40, p. 6669]. The pairing function underlying the definition of a product commutes with concatenations since this is the case for addition and multiplication. Whence if
is stable, then its projection to the first coordinate is stable, and by definition equal to . ∎
We now state the main theorem of this section:
Theorem 3.5.
If is a consequence of ACA, then holds in with truth value .
4 Harvesting the fruits
The aim of this section is threefold. We will spell out some consequences of the transfer principle and interpret these consequences in a standard setting. That is, we will compare a meaning of the same object in and in a standard model. For example, is the set of natural numbers in , while it is the function space of valued functions on in a standard model. If there is no risk of confusion, we will omit a distinction in language between standard and nonstandard objects if the context makes the distinction visible. We will also argue that the transfer principle provides a common framework for many existing results.
We begin by recalling some settheoretical vocabulary in the conditional setting. The inclusion relation [40, Definition II.3.1] on is interpreted by the relation if and only if and , which coincides with the inclusion relation in conditional set theory, see [13, Definition 2.8] for the original definition, and [30, Definition 2.5] for an adaptation to the context of an associated measure algebra (the latter is precisely the above relation).
The product of two sets and is the set by interpreting [40, Definition II.3.1] in . This definition extends the definition of a product in conditional set theory [13, Definition 2.14]. Indeed, a conditional product was introduced in [13] only for sets of the form . It was realized in [30] that the definition given in [13] does not suffice for establishing a conditional version of Fubini’s theorem which relies on the definition of a product of two measure spaces. The definition then used in [30, Section 5.1] coincides with the definition obtained above.
Following [40, Definition II.3.1] a function is thus a subset such that for each there is a unique such that . Due to the same aforementioned reason, this definition extends the one of a conditional function in [13, Definition 2.17] which is an abstraction of earlier concepts named a regular function in [10, Definition 4, Proposition 1], a stable function in [8, Definition 4.2], and a local function in [17, Definition 3.1]. A characteristic property of a function in is that evaluation of its values commutes with concatenations.
From [40, Lemma II.2.1] we know that is a commutative ordered semiring with cancellation. For instance, the totality property reads as the statement that for each pair there is a partition . This interpretation coincides with the definition of conditionally total in [13, Definition 2.15].
A set is said to be finite whenever there exists such that for all , see [40, p. 67], which coincides with the notion of conditionally finite as defined in [13, Definition 2.23]. In general, a finite set in the model is not finite in a standard sense, it might even be uncountable from that perspective. However, for every finite set there exist a partition and a countable family of finite subsets of (in a standard meaning) such that , where we identified with the set of all , cf. [13, Lemma 2.22]. The collection of all finite sequences of length is the set of functions , see [40, Definition II.3.3], which is precisely the construction given in the paragraph after [13, Definition 2.20].
A sequence in a set is a function , compare with the definition of a conditional sequence in [13, Example 2.2.1]. From the standard point of view such a sequence is a net which is parametrized by and commutes with concatenations, see e.g. [15, Section 2].
Let be a sequence of sets in . Using the element relation , the intersection of is
(4.1) 
where
The union of is defined by
(4.2) 
where
The complement of a set is defined in (2.2). The set operations in recover the conditional set operations, see [13, p. 567].
Real analysis and linear algebra
In the abstract system of conditional sets, a detailed construction of the conditional real numbers and its conditional order and topological properties are discussed in [29, Chapter 5] where all properties are proved by hand starting from scratch. In this subsection, we will show that most of these results are consequences of the transfer principle. In the model , the integers (see [40, p. 73]) are the space of integervalued functions on . From [40, Theorem II.4.1], we know that is an ordered integral domain and is Euclidean. Similarly, the space of rationalvalued functions are the rational numbers in . By [40, Theorem II.4.2], is an ordered field, compare with the notion of conditionally ordered field of conditional rational numbers in e.g. [13, Example 4.2.1]. The completion of yields the real numbers which is the space of realvalued measurable functions on , see [40, Definition II.4.4].
Now [40, Theorem II.4.5] implies that is an Archimedian ordered field, a fact which is proved by hand in [29, Lemma 5.2.12, Theorem 5.2.7]. Here, the absolute value takes values in . Let . A open ball (see [40, p. 81]) in is a set
where and . An open set is the union of a sequence of open balls , see [40, Definition II.5.6]. A closed set is the complement of an open set, see [40, Definition II.5.12]. For example, for some , the set is closed, but the standard complement of is not. It can be verified that every closed set is sequentially closed in a standard sense, that is contains the limit of every standard sequence in which converges almost everywhere to . Let denote the set of finite sequences . If , then we have where is a standard product for each . The absolute value on extends to a Euclidean norm on which for can be written as
(4.3) 
Remark 4.1.
In a standard model, the Euclidean norm (4.3) is one example of a “random” or “conditional” or “valued” norm. Such norms appear in different parts such as the analysis of LebesgueBochner spaces (see e.g. [28]), vector integration (see the standard reference [11]), or probabilistic analysis (see e.g. [17, 38]).
By the transfer principle, we obtain from [40, Lemma III.2.1] a BolzanoWeierstraß theorem. Recall that a sequence in is said to be bounded if there exists such that for all .
Theorem 4.2.
Let be a bounded sequence in . Then exists. Moreover, there exists a subsequence , , which converges to .
This theorem is reminiscent of a ‘randomized’ or ‘conditional’ version of the BolzanoWeierstraß theorem as proved with standard techniques and for a standard sequence in a standard product in [33, Lemma 2], in [19, Lemma 1.64] and in [8, Theorem 3.8], respectively.
Compactness within ACA is introduced in [40, Definition III.2.3] as a form of sequential compactness. From this definition we can achieve the following characterization.
Proposition 4.3.
Let be a nonempty subset of closed under countable concatenations. The following are equivalent.

is compact.

HeineBorel property: For every sequence of open sets such that there exists a finite subsequence such that .

where is a partition and is the set of almost everywhere selections of a measurable map with values in the compact subsets of .
Proof.
A conditional version of the HeineBorel theorem was established in [13, Theorem 4.6]. One simple example of a compact set in is the random interval .
The following minimum theorem was proven in conditional analysis in [8, Theorem 4.4], and applied in [6, 31] to stochastic optimal control problems.
Theorem 4.4.
Let be a compact subset of and be a lower semicontinuous function. Then has an infimum.
Proof.
The proof can be done in WKL, which is a subsystem of ACA, using the HeineBorel covering property by [40]. The claim now follows from the transfer principle. ∎
The previous minimum theorem has a variant in setvalued analysis. An interesting aspect of the following result is that it bypasses measurable selection arguments. As for the terminology, one may consult [37, Chapter 14, Sections A and D] and [36, Chapter 1, Sections 1 and 2].
Corollary 4.5.
Let be a normal integrand and be a measurable compactvalued map. Then there exists a measurable function such that almost everywhere.
Proof.
In [8, Section 2], some classical results in linear algebra are extended to culminating in an orthogonal decomposition theorem in [8, Corollary 2.12]. In the standard setting of [8], the space is viewed as an module of rank over the commutative ring . By basic linear algebra in secondorder arithmetic (see [40]), the space is a vector space of dimension in . Thus, all results in [8, Section 2] become consequences of the transfer principle.
Polish, Banach and LebesgueBochner spaces
The aim of this section is show that LebesgueBochner spaces of functions with values in separable spaces correspond to Polish or Banach spaces in . Recall that a Polish space is defined as a completion of a countable set with a prescribed rate of convergence in secondorder arithmetic, see [40, Definition II.5.1].
Let be Polish space with a complete metric and a countable dense subset of in a standard model. The standard metric extends to a metric on with values in by putting almost everywhere. Let denote the subset of consisting of functions with values in . Every can be identified with a sequence in such that implies for all . Indeed, build the conditional sequence , . By a conditional version of the axiom of choice ([13, Theorem 2.26]), we find a conditional sequence such that for all . One concludes that is a Polish space in the sense of [40, Definition II.5.1] in the model . Moreover, if is a standard separable Banach space, then is a separable Banach space in the sense of [40, Definition II.10.1] in , while it is a topological module in a standard setting.
As an illustration of potential results, we obtain a version of Baire category theorem [40, Theorem II.5.8], Urysohn’s lemma [40, Lemma II.7.3], Tietze’s extension theorem [40, Theorem II.7.5], Ascoli lemma [40, Theorem III.2.8], HeineBorel theorem [40, Theorem IV.1.5], a choice principle for points in compact sets [40, Theorem IV.1.8], HahnBanach extension and separation theorems [40, Theorems IV.9.3, X.2.1], BanachSteinhaus theorem [40, Theorem II.10.8], BanachAlouglu theorem [40, Remark X.2.4], and KreinŠmulian theorem [40, Theorem X.2.7] for LebesgueBochner spaces thanks to the transfer principle.
Remark 4.6.
In [14, 15] conditional completions of standard metric spaces are constructed and their connection to LebesgueBochner spaces is discussed. The notion of a random metric can be traced as far back as Menger [35], see [38] for an overview. The notion of a conditional metric space and a conditional normed space are introduced in [13] abstracting earlier concepts in the context of modules or randomly normed spaces in [17, 24, 28]. Conditional vector spaces are introduced in [13, Section 5], and the link to modules is established in [32]. Many of the above listed consequences of the transfer principle were proved by hand with the techniques of conditional analysis, see [32] for an overview and references.
Hilbert spaces
In this subsection, we will elaborate on basic results in separable Hilbert spaces^{5}^{5}5See [1, Definition 9.3] for a definition in secondorder arithmetic. where our focus lies on the following two classes of Hilbert spaces:

the LebesgueBorel space where is a standard separable Hilbert space;

the space
where is a standard Borel probability space, is a subalgebra, and denotes a conditional expectation.
As for the first class, the inner product is defined through parametrization . The inner product in is defined by . Notice that for the base space is .
Let be a Hilbert space in . A sequence in is said to be orthonormal if whenever and . An orthonormal sequence is generating if for every there exists a sequence in such that ^{6}^{6}6Notice that this sum does a priori not have a well defined value in a standard setting as it is uncountable, as a limit in it is though meaningful, and this value can then be interpreted in a standard setting.. A generating orthonormal sequence is called a orthonormal basis. By [1, Theorem 10.9], every separable Hilbert space in has an orthonormal basis. Thus in either of the two aforementioned classes of Hilbert modules there exists an orthonormal basis relative to the linear structure.
From a standard point of view, let be a nonempty subset of a Hilbert module which is stable under countable concatenations, linear and sequentially closed (i.e. one has for all standard sequences in and almost everywhere). Then is a nonempty closed linear subset of the Hilbert space in the model . By the projection theorem [1, Theorem 12.5], every point has a smallest distance . The orthogonal decomposition theorem is a consequence of the projection theorem. We have also a Riesz representation theorem due to [1, Theorem 13.4] which interpreted in a standard setting reads as follows.
Theorem 4.7.
Let be linear and sequentially continuous^{7}^{7}7That is, almost everywhere whenever almost everywhere for standard sequences.. Then there exists such that .
We obtain the following extension of von Neumann’s mean ergodic theorem to conditionally separable Hilbert modules.
Theorem 4.8.
Let be a conditionally separable Hilbert module. Let be linear and such that for all . Then
converges to the projection of to the space of invariant vectors.
Remark 4.9.
The conditional Hilbert space was introduced in [27] motivated by applications in mathematical economics. A Riesz representation theorem and an orthogonal decomposition theorem in complete random inner product spaces are proved in [25, Section 4]. A mean ergodic theorem for complete random inner product spaces is established in [26].
Fixed point theorems
A conditional version of the Brouwer fixed point theorem is proved in [16]. The precise statement in the standard model is the following.
Theorem 4.10.
Let be finitely many points in for . Let
Let be a local and sequentially continuous function. Then there is such that .
The proof is based on an adaptation of Sperner’s lemma to a conditional setting. One obtains Sperner’s lemma and the Brouwer fixed point theorem in the following slightly more general form as a consequence of a transfer principle applied to [40, Theorem IV.7.6]. The Brouwer fixed point theorem in is the following statement.
Theorem 4.11.
Let and be a finite sequence of points in . Let denote the simplex generated by . Then every continuous function has a fixed point.
Moreover, we have the MarkovKakutani fixed point theorem by applying the transfer principle to [40, Lemma IV.9.1].
Theorem 4.12.
Let be closed convex set in the sequence space where is the compact interval in . Let be a sequence of affine continuous functions such that for all and . Then there is such that for all .
Further connections and outlook
We have left out measure theory. For the development of measure theory in secondorder arithmetic, we refer to [39, 40, 42]. In conditional set theory, basic results in measure theory are established in [30] in a standard setting of stable spaces of functions on . As measure theory in secondorder arithmetic is based on Daniell’s functional approach to integration, the conditional version of the DaniellStone and Riesz representation theorems in [30, Section 5] can be connected to respective theorems in , but a conditional version of the RadonNikodým theorem (see [30, Section 5]) as well. In [30, Section 4], a connection between kernels in a standard model and measures in conditional set theory are proved. This connection fully carries over to the model , for Borel probability measures on where is a standard compact metric space.
References
 [1] J. Avigad and K. Simic. Fundamental notions of analysis in subsystems of secondorder arithmetic. Ann. Pure Appl. Logic, 139:138–184, 2006.
 [2] A. Aviles and J. M. Zapata Booleanvalued Models as a Foundation for Locally Convex Analysis and Conditional Set Theory. IFCoLog Journal of Logic and its Applications, 5(1):1–31, 2018.
 [3] J. Backhoff and U. Horst. Conditional analysis and a PrincipalAgent problem. SIAM J. Financial Math., 7(1):477–507, 2016.
 [4] J. L. Bell. Set Theory: BooleanValued Models and Independence Proofs. Oxford Logic Guides. Clarendon Press, 2005.
 [5] T. Bielecki, I. Cialenco, S. Drapeau, and M. Karliczek. Dynamic Assessement Indices. Stochastics, 88(1):1–44, 2016.
 [6] P. Cheridito, U. Horst, M. Kupper, and T. Pirvu. Equilibrium pricing in incomplete markets under translation invariant preferences. Math. Oper. Res., 41(1):174–195, 2016.
 [7] P. Cheridito and Y. Hu. Optimal consumption and investment in incomplete markets with general constraints. Stoch. Dyn., 11(2):283–299, 2011.
 [8] P. Cheridito, M. Kupper, and N. Vogelpoth. Conditional analysis on . Set Optimization and Applications, Proceedings in Mathematics & Statistics, 151:179–211, 2015.
 [9] P. Cheridito and M. Stadje. BSEs and BSDEs with nonLipschitz drivers: comparison, convergence and robustness. Bernoulli, 19(3):1047–1085, 2013.
 [10] K. Detlefsen and G. Scandolo. Conditional and Dynamic Convex Risk Measure. Finance Stoch., 9:539–561, 2005.
 [11] J. Diestel and J. J. Uhl. Vector Measures. Mathematical surveys and monographs. American Mathematical Society, 1977.
 [12] S. Drapeau and A. Jamneshan. Conditional preferences and their numerical representations. J. Math. Econom., 63:106–118, 2016.
 [13] S. Drapeau, A. Jamneshan, M. Karliczek, and M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl., 437(1):561– 589, 2016.
 [14] S. Drapeau, A. Jamneshan, and M. Kupper. Vector duality via conditional extension of dual pairs. Preprint available at arXiv:1608.08709, 2016.
 [15] S. Drapeau, A. Jamneshan, and M. Kupper. A FenchelMoreau theorem for valued functions. Preprint available at arXiv:1708.03127, 2017.
 [16] S. Drapeau, M. Karliczek, M. Kupper, and M. Streckfuss. Brouwer fixed point theorem in . J. Fixed Point Theory Appl., 301(1), 2013.
 [17] D. Filipović, M. Kupper, and N. Vogelpoth. Separation and duality in locally convex modules. J. Funct. Anal., 256:3996–4029, 2009.
 [18] D. Filipović, M. Kupper, and N. Vogelpoth. Approaches to conditional risk. SIAM J. Financial Math., 3(1):402–432, 2012.
 [19] H. Föllmer and A. Schied. Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics. Walter de Gruyter, Berlin, New York, 3 edition, 2011.
 [20] D. Fremlin. Topological Riesz Spaces and Measure Theory. Cambridge University Press, 1974.
 [21] M. Frittelli and M. Maggis. Complete duality for quasiconvex dynamic risk measures on modules of the type. Stat. Risk Model., 31(1):103–128, 2014.
 [22] S. R. Givant and P. R. Halmos. Introduction to Boolean Algebras. Undergraduate texts in mathematics. Springer, 2009.
 [23] S.M. Grad and A. Jamneshan. Graphtype regularity conditions for strong duality in a vector optimization context. In preparation.
 [24] T. Guo. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal., 258:3024–3047, 2010.
 [25] T. Guo. On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules. J. Funct. Spaces, pages 1–13, 2013.
 [26] T. Guo and X. Zhang. Von Neumann’s mean ergodic theorem on complete random inner product modules. Front. Math. China, 6(5):965–985, 2011.
 [27] L. P. Hansen and S. F. Richard. The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models. Econometrica, 55:587–613, 1987.
 [28] R. Haydon, M. Levy, and Y. Raynaud. Randomly Normed Spaces. Hermann, 1991.
 [29] A. Jamneshan. A Theory of Conditional Sets. PhD thesis, HumboldtUniversität zu Berlin, 2014.
 [30] A. Jamneshan, M. Kupper, and M. Streckfuß. Measures and integrals in conditional set theory. SetValued Var. Anal., to appear.
 [31] A. Jamneshan, M. Kupper, and J. M. Zapata. Parameterdependent stochastic optimal control in finite discrete time. Preprint available at arXiv: 1705.02374, 2017.
 [32] A. Jamneshan and J. M. Zapata. On compactness in modules. Preprint available at arXiv:1711.09785, 2017.
 [33] Y. Kabanov and C. Stricker. A teacher’s note on noarbitrage criteria. Séminaire de Probabilités, 35:149–152, 2001.
 [34] A. Kechris. Classical Descriptive Set Theory. Graduate Texts in Mathematics. SpringerVerlag, 1995.
 [35] K. Menger. Statistical Metrics. Proc. Natl. Acad. Sci. USA, 28(12):535–537, 1942.
 [36] I. Molchanov. Theory of Random Sets. Probability and Its Applications. Springer London, 2005.
 [37] R. T. Rockafellar and R. J.B. Wets. Variational Analysis. Springer, Berlin, New York, 1998.
 [38] B. Schweizer and A. Sklar. Probabilistic Metric Spaces. Dover Publications, 2011.
 [39] K. Simic. Aspects of Ergodic Theory in Subsystems of Secondorder Arithmetic. PhD thesis, Carnegie Mellon University, 2004.
 [40] S. G. Simpson. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. SpringerVerlag Berlin Heidelberg, 1999.
 [41] G. Takeuti. Proof Theory. Dover books on mathematics. Dover Publications, Incorporated, 2013.
 [42] X. Yu. Riesz representation theorem, Borel measures and subsystems of secondorder arithmetic. Ann. Pure Appl. Logic, 59:65–78, 1993.