A transfer principle for second-order arithmetic, and applications

In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics 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 second-order 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, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}.This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model.

In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics 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 second-order 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, set-theoretical topology or uncountable set theory, see e.g. the introduction of Simpson [47]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model. 03C90,03F35; 28B20,54C65

Introduction
Fixing a probability space (Ω, F, P), one can distinguish between probabilistic and deterministic objects such as a deterministic real number which is an element of R and a random real number which is a measurable function x : Ω → R. Extending such reasoning, one might speak of a 'random', 'stochastic', 'measurable' or 'conditional' version of a 'deterministic', 'classical' or 'standard' theorem which expresses a randomization of its statement. To illustrate, a conditional version of the Bolzano-Weierstraß theorem states that for every sequence (x k ) of random real numbers such that x = lim sup x k < ∞ almost surely, there exists a strictly increasing sequence n 1 < n 2 < . . . of integer-valued random variables such that x n k converges almost surely to x. Experience has shown that many classical theorems have such conditional analogues such as the Heine-Borel theorem, the Hahn-Banach extension and separation theorems and the Brouwer fixed point theorem, see e.g. Cheridito et al. [10], Drapeau et al. [15,18], Filipovic et al. [19], and Jamneshan et al. [36,38] for an account. 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 second-order logic. Strong arguments have been put forward in favor of the claim that second-order logic is a satisfying formal framework for the bulk of classical mathematics, see e.g. the introduction to Simpson [47]; 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 0 , which has a secondorder comprehension axiom for formulas in which all quantifiers range over natural numbers (see Simpson [47]), also holds conditionally. To this end, we show (i) that the axioms of ACA 0 hold in the structure with first order part L 0 (N) and second-order part its conditional power set P with truth value Ω, and (ii) that truth value Ω is preserved by the usual deduction rules of second-order predicate calculus. To this end, a conditional element relation between L 0 (N) and P is introduced. We expect that, with a certain amount of extra technical effort, this can be extended to a considerably stronger transfer principle for full second-order logic.
In section 4, we then discuss consequences of the transfer principle. We verify that the transfer principle yields a conditional version of a whole class of classical theorems many of which were proved by hand previously 1 . As new consequences, we obtain a conditional version of the Peano existence theorem, Urysohn's lemma, the existence of an orthonormal basis and the Markov-Kakutani theorem.
A conditional version of a classical theorem is oftentimes also a theorem about a more involved situation in a classical setting. For instance, the above conditional version of the Bolzano-Weierstraß-theorem is also a classical theorem securing existence of measurably parametrized almost surely converging subsequences of an almost surely upper bounded sequence of real-valued measurable functions 2 . We will systematically investigate a standard interpretation of a conditional version of a classical structure or 1 The latter practice provides some useful insight into the transfer process. 2 In this classical form under the name of a 'randomized', 'stochastic' or 'measurable' version of the Bolzano-Weierstraß-theorem this statement is proved in Föllmer and Schied [21] and Kabanov and Stricker [39], motivated by applications in mathematical economics. theorem in section 4. As a result, bridges to the analysis in L 0 (R)-modules (e.g. Cheredito et al. [10], Filipovic et al. [19]) and set-valued analysis (e.g. Rockafellar and Wets [45]) are built.
We will be also interested in the reverse direction. Namely, it can be verified that certain classical structures have a useful interpretation in a conditional model. For example, the standard space L 0 (R) of real-valued Borel measurable functions on a σ -finite measure space (Ω, F, µ) modulo almost everywhere equality are the real numbers in a conditional model. More generally, if E is a complete separable metric space and L 0 (E) is the space of E -valued Borel measurable functions on Ω modulo almost everywhere equality, we show that L 0 (E) can be identified with a complete separable metric space in a conditional model of ACA 0 . The space L 0 (E) reflects a measurable parametrization of the elements of E relative to a base space (Ω, F, µ). Such a measurable parametrization is a constituent part of a conditional model 3 . 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 preserves measurability, and thus provides an alternative to uniformization theorems in descriptive set theory (see e.g. Kechris [40], Molchanov [44], or Rockafellar and Wets [45]), whenever one restricts attention to almost everywhere Borel selections. For example, we show that a maximum theorem in a conditional model is equivalent to a maximum theorem for normal integrands. In particular, compact subsets of a Euclidean space in a conditional model are uniquely related to compact-valued maps.
Conditional set theory (Drapeau et al. [15]), which is used to build a 'conditional' 3 Therefore conditional models might equally be called 'stochastic' or 'measurable' models of an axiomatic system. model of ACA 0 , is conceptually closely related to Boolean-valued models and topoi of sheaves, see Jamneshan [35]. In fact, a conditional model of ACA 0 is a Booleanvalued model of ACA 0 by changing to the measure algebra associated to an underlying base space (Ω, F, µ). The constructive approach of second-order arithmetic permits us to explore the semantics in a Boolean-valued model, and through this understanding build a relationship to structures and theorems in a standard model, and thus facilitate applications.
In Aviles and Zapata [2], a categorical equivalence between conditional sets and a certain class of Boolean-valued sets is discussed. The correspondence is based on ZFC set theory rather than second-order arithmetic. We believe that our transfer principle is of independent interest, as (1) it is an explicitly formulated theorem; (2) second-order 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, Theorem 3.1] excludes local subsets which are necessary to prove a transfer principle as shown in the proof of theorem 3.4 below.
The remainder of this paper is organized as follows. In section 2, we set up the stage and collect relevant notions from conditional set theory. In section 3, we prove a transfer principle for ACA 0 . In section 4, we discuss consequences and the applicability of the transfer principle. We conclude by a discussion of potential extensions of the transfer principle in section 5.

Preliminaries
In the language L 2 of second-order arithmetic (see Simpson [47] for an introduction), we distinguish between number variables, traditionally written in lower-case Latin letters x, y, z, ... and set variables, usually written as upper-case Latin letters X, Y, Z, .... Moreover, we have two constant symbols 0 and 1, two binary function symbols + and · and a binary relation symbol <. Between numbers and sets exists an element relation ∈. The first-order terms are number variables, constant symbols and expressions of the form t 1 + t 2 and t 1 · t 2 for first-oder terms t 1 and t 2 . The atomic formulas are t 1 = t 2 , t 1 < t 2 and t 1 ∈ X where t 1 and t 2 are first-order terms and X 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 0 ) consists of the axioms for discretely ordered semirings, together with the second-order induction principle and an axiomatic scheme saying that for any second-order formula φ containing only first-order quantifiers there is a set of all natural numbers of which φ holds, see [47]. By the cumulative results of reverse mathematics, ACA 0 is a sufficient axiomatic basis for a great number of theorems from classical mathematics; many examples can be found in [47].
Throughout fix a σ -finite measure space (Ω, F, µ), and let N denote its σ -ideal of null sets. We always identify A, B ∈ F whenever A∆B ∈ N where ∆ denotes symmetric difference. The resulting quotient Boolean algebra has the following relevant properties: • Completeness: Any family in F has a union and an intersection in F ; • Countable chain condition: Any pairwise disjoint family in F is at most countable; see e.g. Givant and Halmos [25, Chapter 31] for a reference. We will always identify two functions x and y on Ω with the same codomain if {ω : x(ω) = y(ω)} ∈ N . For a function x on Ω and A ∈ F , we write x|A for the restriction of x to A. For a Polish space E , let L 0 (E) denote the space of Borel functions x : Ω → E . In particular, L 0 (R) denotes the space of real-valued measurable functions, and L 0 (N), L 0 (Z) and L 0 (Q) denote its subsets of functions with values in N = {0, 1, . . .}, the integers Z and the rational numbers Q respectively. Recall that L 0 (R) is a Dedekind complete Riesz lattice where addition, multiplication and order are defined pointwise, see e.g. Fremlin [22] for a reference. All inequalities between real-valued measurable functions shall be understood in the almost everywhere sense. By an abuse of language, we also denote by 0 and 1 the functions with constant values 0 and 1 respectively. For a measurable partition (A k ) and a countable family (x k ) in L 0 (E) for some Polish space E , we write k x k |A k for the unique element x ∈ L 0 (E) such that x|A k = x k |A k for all k. We introduce the conditional power set of L 0 (N), see Drapeau et al. [15] for an introduction to conditional set theory. We define a concatenation of a countable family (N k |B k ) in P and a measurable partition (A k ) by where k N k |A k := k n k |A k : n k ∈ N k for each k Let us define conditional intersection and conditional complement. We follow the presentation in Jamneshan et al. [36,Section 2]. Let N|A, M|B ∈ P . The conditional intersection of N|A and M|B is defined as The conditional complement of N|A is the conditional subset By applying an exhaustion argument, it can be derived from stability that C and D are attained, and it can be checked that V and W are stable sets as well. We conclude that the conditional intersection and conditional complement are well defined, see e.g. [36] for a complete argument.
We introduce a conditional element relation between L 0 (N) and its conditional power set P which is a new ingredient in conditional set theory.
for k 2 which defines a measurable partition of A * . Since A k ∈ G for all k, the claim follows from stability of N . Further, it can be directly checked that Our aim is to prove that every consequence of ACA 0 holds in the structure S := (L 0 (N), P, +, ·, 0, 1, <, i) with truth value Ω, i.e. that S is a 'conditional model' of ACA 0 . We start by explaining the evaluation of terms and formulas in S .
Let V 1 denote the collection of all number variables and let V 2 denote the collection of all set variables. Let β be a function with domain V 1 ∪ V 2 such that β : V 1 → L 0 (N) and β : V 2 → P , and let t be a first-order term. Such a β is called a 'conditional assignment'. Then [t] β , the β -evaluation of t, is defined recursively as For first-order terms t 0 and t 1 and a second-order variable X , the conditional βevaluation of atomic formulas is defined by The conditional β -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 Boolean-valued models, see e.g. [4, Chapter 1].
Proposition 3.1 Let β be a conditional assignment and let φ and ψ be formulas in L 2 . Then there exist n ∈ L 0 (N) and N|A ∈ P such that Proof We may assume that [∃xφ(x)] β = Ω. We find a countable family The second claim can be shown analogously by using concatenations in P .
We will now adapt the usual notion of the correctness of a sequent to the conditional context. Definition 3.2 If Γ and ∆ are sets of second-order formulas, then Γ → ∆ is called a sequent. The conditional validity of a sequent Γ → ∆ with respect to a conditional assignment β is defined by and Γ → ∆ is said to be correct if and only if for all assignments β . An inference rule R is a pair consisting of a finite sequence (Γ 1 , Γ 2 , . . . , Γ n ) of sequents and a single sequent Γ written as Γ 1 ,Γ 2 ,...,Γn Γ , and it is said to be correct if and only if the correctness of Γ follows from the correctness of We want to apply the inference rules for second-order logic given by Takeuti in [50, p. 9-10 and p. 135-136]. By Boolean arithmetic, one can directly check that all structural and logical rules are correct. We illustrate this for the first weakening rule which leads to the correctness of (3-1). By Proposition 3.1, the left universal quantification rule where t is a term, is correct. As for the correctness of the right universal quantification rule where y does not occur freely in Γ → ∆, ∀xφ(x), assume for all conditional assignments and let β be an arbitrary conditional assignment. We want to show that . Choose a y-variant β ′ of β that maps y to n * . Then (3-3) follows from (3-2) since y does not occur freely in Γ → ∆, ∀xφ(x), so that the first two sets of the union (3-3) remain unchanged. Analogously, one shows the first-order existential quantification inference rules. The second-order quantifier inference rules are only relevant for second-order variables, as predicate constants do not appear in our language. The inference rules for second-order quantification can hence be proved analogously to the corresponding first-order rules. Thus, we obtain: is any deduction rule of second-order sequent calculus and Γ 1 , ..., Γ n are correct, then Γ is correct. In particular, if all elements of Γ hold in S with truth value Ω and Γ → φ is derivable with the rules of sequent calculus, then φ holds in S with truth value Ω.
Theorem 3.4 All the axioms of ACA 0 attain the value Ω in the structure S for all conditional assignments.
Proof Let β be an arbitrary conditional assignment. The verification of the basic axioms (see Simpson [47, p. 4]) is immediate from the definitions. For the sake of completeness, we provide the elementary arguments below.
As for the second-order induction scheme, we have to verify that By conditionally evaluating the previous formula and rewriting it by using Boolean arithmetic, we must verify that A ⊂ B, where But this is immediate from the stability of β(X).
Finally, we verify the arithmetical comprehension scheme, that is, we want to show that [∃X∀x(x ∈ X ↔ φ(x))] β = Ω for any arithmetical 4 formula φ(x) in which X does not occur freely. By Proposition 3.1, for all n ∈ L 0 (N). Indeed, for n 0 ∈ L 0 (N), let n 1 ∈ N φ be such that which proves comprehension. Thus it remains to verify that N φ is stable under concatenations for all formulas φ, which we will prove by an induction on arithmetical formulas. First, since addition and multiplication commute with concatenations 5 , for any first-order term t = t(x), by an induction on terms, one has for all measurable partitions (A k ) and every countable family (n k ) in L 0 (N). Since also order and concatenations commute and due to (2)(3), N φ is stable for all atomic formulas φ.
Let φ and ψ be two arithmetical formulas such that N φ |A φ , N ψ |A ψ ∈ P . Then we have Moreover, for a negation one obtains Finally, let θ(x, y) be arithmetical and φ(y) = ∃xθ(x, y). Clearly, A φ = A θ . By the established, we can already define a pairing function, product of stable sets and their projections, see e.g. Simpson [47, p. 66-69]. The pairing function (i, j) → (i + j) 2 + i 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 N φ .
We now state the main theorem of this section.
Theorem 3.5 If φ is a consequence of ACA 0 , then φ holds in S with truth value Ω.
Proof By theorem 3.4, all axioms of ACA 0 have truth value Ω in S . By Lemma 3.3, truth value Ω is preserved under the rules of second-order sequent calculus. Thus, the theorem follows.

Harvesting the fruits
A first step towards applications of the model-theoretic results in this article is to investigate connections between a conditional and a standard model. An aim of this section is to understand consequences of the transfer principle and interpret them properly in a standard setting. For example, the structure L 0 (N) are the natural numbers in the 'conditional model', while interpreted in a standard model, it is the space of N-valued measurable functions on a σ -finite measure space. We shall discover connections to different parts of analysis such as set-valued analysis and measurable selection theory, vector-valued analysis and probabilistic analysis with applications to areas such as stochastic control, vector optimization and mathematical economics. An important observation is that application of theorems in a 'conditional model' preserves measurability systematically by construction. Moreover, we will demonstrate that a transfer principle unifies a conditional version of classical theorems which were proved by hand previously. Finally, new examples of theorems and applications shall underline the usefulness of a transfer principle.

Set theory
We begin by collecting basic set-theoretical vocabulary in a conditional setting. The The product of two sets N|A, M|B ∈ P is the set  Let (N k |A k ) be a sequence in the power set P . Using the element relation (see Definition 2.2), we see that the intersection of (N k |A k ) can be identified with and that the union of (N k |A k ) can be identified with The complement of a set was defined in (2-2). We have seen that the set operations in S recover the conditional set operations [15, p. 567]. In [15,Corollary 2.10], it was proved that the conditional power set has the structure of a complete Boolean algebra, which is a fundamental result for basic constructions in conditional topology [15,Section 3] and conditional measure theory [36]. The transfer principle for ACA 0 implies a weaker statement, namely that the power set P has the structure of a σ -complete Boolean algebra, see [47].

Real analysis and linear algebra
A detailed construction of the conditional real numbers and their conditional algebraic, order and topological properties are developed in Jamneshan [34,Chapter 5] where all properties are proved from conditional set theory. We will argue that most of these properties and related results are consequences of the transfer principle. of B(0, r) is in general not. However, it can be easily verified that every S -closed set is sequentially closed, i.e. it contains the limit of any almost everywhere converging standard sequences.

Remark 4.3
The Euclidean topology of the real numbers L 0 (R) in S is the order topology in a standard model which renders pointwise addition and multiplication continuous. Thus, L 0 (R) becomes a topological algebra, in particular a topological module of rank 1 over itself. This fact is exploited in Filipovic et al. [19], Cheridito et al. [10], Drapeau et al. [15], Jamneshan and Zapata [38] to build a functional analytic discourse in L 0 (R)-modules. We shall observe later that a functional analytic discourse in L 0 (R)-modules is the reflection of classical functional analysis, albeit in a 'conditional model'. So far, we know that L 0 (R) is 1-dimensional Euclidean space in the 'conditional model' S .
Let L 0 (R) n denote the set of finite sequences {1 m n} → L 0 (R). Write n as k n k |A k for (n k ) in N and (A k ) a measurable partition. Then one can view L 0 (R) n as k L 0 (R) n k |A k where each L 0 (R) n k is a standard product. The absolute value extends from L 0 (R) to an Euclidean norm on L 0 (R) n which for x = k (x 1 , . . . , x n k )|A k is defined by (4)(5) x := k x 2 1 + . . . + x 2 n k A k Convergence in the conditional Euclidean space L 0 (R) n can be characterized by almost everywhere convergence. Suppose that n ∈ N (the general case n = k n k |A k follows by localizing the subsequent construction to each A k and gluing). Let (x k ) be a standard sequence in L 0 (R) n converging almost everywhere to x. Given r ∈ L 0 ++ (Q), let Notice that k r is measurable. Obtain from (x k ) the conditional sequence x k := n x kn |A n , k = n k n |A n . By construction, x k − x < r for all k k r . Conversely, it is easy to see that if (x k ) k∈L 0 (N) is a conditional sequence conditionally converging to x, then the standard subnet (x k ) k∈N resulting by embedding N into L 0 (N) via n → n1 Ω converges to x almost everywhere.
By the transfer principle, we obtain from [47, Lemma III.2.1] a Bolzano-Weierstraß theorem in S . Recall that a sequence (x k ) in L 0 (R) n , n ∈ L 0 (N), is said to be bounded if there exists r ∈ L 0 ++ (Q) such that x k < r for all k ∈ L 0 (N). Proposition 4.6 Let W be a stable subset of L 0 (R) n , n = k n k |A k ∈ L 0 (N). Then the following are equivalent.
(i) W is conditionally compact.
(ii) Heine-Borel property: For every conditional sequence ( (iii) W can be represented as k W k |A k where each W k is the set of almost everywhere selections of an Effros measurable 6 compact-valued map in R n k .  where a < b in L 0 (R).
The following minimum theorem was proved in [10,Theorem 4.4] in finite dimensions, and in full generality in Jamneshan and Zapata [38,Theorem 5.13], and applied in e.g. Cheridito et al. [8] and Jamneshan et al. [37] successfully to stochastic control.
Theorem 4.8 Let W be a conditionally compact subset of L 0 (R) n , n ∈ L 0 (N), and let f : W → L 0 (R) be a conditionally lower semi-continuous function. Then f has a minimum.
Proof The proof can be done in WKL 0 which is a subsystem of ACA 0 using the Heine-Borel covering property [47]. The claim then follows from theorem 3.5.
We can derive the following variant of the previous theorem in set-valued analysis an important aspect of which is the avoidance of measurable selection arguments.
Corollary 4.9 Let f : Ω × R n → R be a normal integrand 7 , and let X : Ω → 2 R n be an Effros measurable compact-valued map. Then there exists a measurable function x : Ω → R n such that f (ω, x(ω)) = min x∈X(ω) f (ω, x) almost everywhere.
Proof By proposition 4.6, one can identify X with a conditionally compact set in L 0 (R) n . From a result in Jamneshan et al. [37,Section 5], one can identify f with a sequentially lower semi-continuous 8 function L 0 (R) n → L 0 (R). Theorem 4.8 proves the claim.
A notion of an L 0 (R)-derivative for functions f : L 0 (R) n → L 0 (R) is introduced in Cheridito et al. [10,Section 7]. An interpretation of the definition of a derivative in second-order arithmetic yields the same concept which is defined below for n = 1.
The following conditional version of the Peano existence theorem, derived from [47, Theorem IV.8.1], has not been proved previously. It can be applied to solve random ordinary differential equations. In [10, Section 2], some basic results in linear algebra are extended to the space L 0 (R) n which culminates in a conditional version of the orthogonal decomposition theorem [10, Corollary 2.12], whereby L 0 (R) n is viewed as an module of rank n over the commutative ring L 0 (R). By basic linear algebra in second-order arithmetic [47], L 0 (R) n is a vector space of dimension n in S , and all results in [10, Section 2] are consequences of the transfer principle. Actually, they directly extend to the case where n(ω) is a measurable dimension.

Metric spaces
In the axiomatic system of second-order arithmetic, the only definable metric spaces are separable and complete ones which are coded as a completion of a countable set with a prescribed rate of convergence, see Simpson [47, Definition II.5.1]. We characterize a complete and separable metric space in the conditional model S as a vector metric space in a standard model as follows.
Definition 4.13 A non-empty set H is said to be a conditional set 9 , if there exists a restriction operation | such that for every sequence (x k ) in H and every measurable partition (A k ) there exists a unique element x ∈ H such x k |A k = x|A k for all k. We name this unique element a concatenation and denote it by x = k x k |A k .
Let H be a conditional set. A function d : A conditional metric space H is said to be • conditionally separable, if there exists a countable set G ⊂ H such that for all x ∈ H there is a sequence (x k ) in G and n 1 < n 2 < . . . in L 0 (N) such that d(x nm , x) → 0 almost everywhere where x n := k x k |{n = k} for n ∈ L 0 (N); • conditionally complete, if for every conditional Cauchy sequence 10 (x k ) there exists x such that d(x k , x) → 0 almost everywhere.
A conditional metric space is said to be conditionally Polish, if it is conditionally separable and conditionally complete.    We prove a characterization of closed sets in S which establishes a link to set-valued analysis.
Proposition 4.17 Let E be a standard Polish space, and let W ⊂ L 0 (E) be stable 11 . Then the following are equivalent.
(i) W is closed in S .
(ii) W is sequentially closed.
(iii) There exists an Effros measurable and closed-valued map X : Ω → 2 E such that W coincides with the set of almost everywhere selections of X .
Let (x k ) be a sequence in W such that d(x k , x) → 0 almost everywhere for some x ∈ L 0 (E), and let G be a countable dense set in E . By contradiction, if µ(i(x, W ⊏ )) > 0, then one finds a conditional ball B(y, r) with center y ∈ L 0 (G) and radius r ∈ L 0 ++ (Q) such that µ(i(x, B(y, r) ⊓ W ⊏ )) > 0. But then there is also some k such that µ(i(x k , W ⊏ )) > 0 which is the desired contradiction. Hence µ(i(x, W ⊏ )) = 0, and therefore x ∈ W .
As a direct consequence of the transfer principle, one obtains a conditional version of the

Banach space theory
In this section, we develop basics of Banach spaces in S , and connect it with functional analysis in L 0 (R)-modules. An L 0 (R)-module H is a module over the commutative algebra L 0 (R). This yields a function from the underlying measure space (Ω, F, µ) to H by scalar multiplication with indicator functions 1 A · x, A ∈ F and x ∈ H 12 . This enables to formalize concatenations in H : An element x ∈ H is said to be a concatenation of a sequence (x k ) in H and a measurable partition (A k ), if 1 A k · x = 1 A k · x k for all k. We say that H satisfies the countable concatenation property, if for every sequence (x k ) in H and each measurable partition (A k ) there exists a unique concatenation in H . Throughout all L 0 (R)-modules are assumed to satisfy the countable concatenation property.   Notice that any conditionally linear function is stable. One can deduce from [47,Theorem II.10.7] that a conditionally linear and conditionally bounded operator is sequentially continuous, i.e. f (x k ) − f (x) K → 0 almost everywhere whenever x k − x H → 0 almost everywhere. A conditional version of the Banach-Steinhaus theorem is obtained by an application of the transfer principle to [47,Theorem II.10.8]. One could think of its application in random operator theory and random differential equations, see e.g. Skorohod [48] and Strand [49].
Theorem 4.25 Let H and K be conditionally separable and complete L 0 (R)-normed modules, and let (f n ) be a standard sequence of L 0 (R)-linear and sequentially continuous functions f n : H → K . If for every x ∈ H there exists r ∈ L 0 ++ (Q) such that f n (x) K r for all n, then there exists q ∈ L 0 ++ (Q) such that f n (x) K q x H for all x and every n.  [20] and Frittelli and Maggis [24] in risk measure theory. A conditional Fenchel-Moreau theorem is applied in Drapeau et al. [17] in vector duality. A conditional Riesz representation theorem is used in Drapeau and Jamneshan [14] in decision theory. Remark 4.27 Separation and duality results in topological L 0 (R)-modules are established in Guo et al. [29] and Filipovic et al. [19] with respect to two types of module topologies respectively, their connection is discussed in Guo [27], see also the discussion in Jamneshan and Zapata [38]. The standard topologies employed in [29], and with a different motivation in Haydon et al. [32], are extensions of the topology of convergence in probability to L 0 (R)-vector norms on general L 0 (R)-modules. The class of such probabilistic topologies does not yield a comprehensive functional analytic discourse in L 0 (R)-modules, see Jamneshan and Zapata [38].
The second type of topologies introduced in [19] proved to be more susceptible to a comprehensive functional analytic discourse for which strong evidence was provided in [15] by embedding L 0 (R)-module theory in conditional set theory and conditional topology. Conditional topological vector spaces are introduced in [15, Section 5]. In Drapeau et al. [16] conditional completions of standard metric spaces are constructed and their connection to Lebesgue-Bochner spaces is established. The above listed consequences of the transfer principle have been proved by hand previously, see [38] for an overview and references.
A school of Russian mathematicians, starting with Kantorovic, studied to which extent results in functional analysis remain true if the real numbers are replaced by a Dedekind complete vector lattice of which L 0 (R) is one example. These investigations were naturally connected to Boolean-valued models. We refer to Kusraev and Kutateladze [42] and Kutateladze [43] for an extensive overview of this tradition.

Hilbert spaces
In this subsection, we will elaborate on basic results in separable Hilbert spaces 14 where our focus lies on the conditional L 2 -space where (Ω, F, P) is a standard Borel probability space and G ⊂ F is a sub-σ -algebra, see examples 4.14. The inner product in L 2 (F|G) is defined by x, y := E[xy|G].
Recall that for L 2 (F|G) the base space is (Ω, G, P). An orthonormal sequence (x n ) n∈L 0 (N) is generating if for every x ∈ H there exists a sequence (r n ) n∈L 0 (N) in L 0 (G, R) such that x = n r n x n 15 . A generating orthonormal sequence is called an orthonormal basis.
The transfer principle applied to Avigad and Simic [1, Theorem 10.9] yields the following new result. Theorem 4.30 L 2 (F|G) has an orthonormal basis in S . Remark 4.31 The significance of theorem 4.30 lies in the fact that generally an L 0 (R)module with the countable concatenation property has an algebraic basis if and only if it is finitely ranked, see Jamneshan and Zapata [38,Proposition 3.5]. This means that an infinite dimensional space such as L 2 (F|G) does not have any module linear basis in a standard sense in general 16 . The difference in S is that one allows the family which forms the basis to be a conditional family, cf. [38, Section 3].
We have the following projection theorem thanks to the transfer principle and [1, Theorem 12.5].
Theorem 4.32 Let W be a sequentially closed sub-module of L 2 (F|G). Then every point x ∈ L 2 (F|G) has a smallest distance d(x, W) := inf y∈W d(x, y).
A consequence of the projection theorem is the orthogonal decomposition theorem: Theorem 4.33 Let W be a sequentially closed sub-module of L 2 (F|G). Then there exists a sequentially closed sub-module W ⊥ such that each element x ∈ L 2 (F|G) has a unique decomposition x = z + y where z ∈ W and y ∈ W ⊥ .
We have also a Riesz representation theorem due to [1,Theorem 13.4] which interpreted in a standard setting reads as follows. We obtain the following extension of von Neumann's mean ergodic theorem, see [1] for the result in ACA 0 .
Theorem 4.35 Let T : L 2 (F|G) → L 2 (F|G) be L 0 (R)-linear such that Tx x for all x ∈ L 2 (F|G). Then 1/n(x + Tx + . . . + T n−1 x) converges in the vector norm of L 2 (F|G) almost everywhere to the projection of x to the sequentially closed sub-module of all T -invariant vectors. Remark 4.36 The conditional Hilbert space L 2 (F|G) was introduced in Hansen and Richard [31] for purposes of financial modeling. Some applications in stochastic analysis of an orthogonal decomposition result are described in Cerreia-Vioglio et al. [7]. A Riesz representation theorem, a projection theorem and an orthogonal decomposition theorem in complete random inner product spaces are proved in Guo [28,Section 4]. A mean ergodic theorem for complete random inner product spaces is established in Guo and Zhang [30].

Fixed point theorems
We close this section with fixed point theorems in the 'conditional model' S . One could think of applications in equilibrium theory and random differential equations.
A conditional version of the Brouwer fixed point theorem was proved in Drapeau et al. [18]. The precise statement in the standard model is the following. The proof in [18] 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 the transfer principle applied to Simpson [47, Theorem IV.7.6].
Theorem 4.39 Let k, n ∈ L 0 (N) and (x i ) i k be a conditionally finite sequence of points in L 0 (R) n . Without loss of generality suppose that k = m k m |A m and n = m n m |A m , and let (x m i ) i km be finitely many elements in L 0 (R) nm which comprise the conditional sequence (x i ) i k on A m for each m. Build the conditional simplex Let S be a stable, sequentially closed and L 0 (R)-convex 18 subset of [−1, 1] L 0 (N) . Let (f k ) be a conditional sequence 19 of L 0 (R)-affine and sequentially continuous functions f k : S → S such that f k • f m = f m • f k for all m, k ∈ L 0 (N). Then there exists x ∈ S such that f k (x) = x for all k.

Further connections and outlook
We have left out measure theory. For the development of measure theory in secondorder arithmetic, we refer to Simic [46], Simpson [47] and Yu [52]. In conditional set theory, basic results in measure theory are established in Jamneshan et al. [36]. As measure theory in second-order arithmetic is based on Daniell's functional approach to integration, the conditional version of the Daniell-Stone theorem and Riesz representation theorem in [36,Section 5] can be connected to respective theorems in S . In For some mathematical theorems relevant for applications in mathematical economics, stronger second-order axioms than ACA 0 are required. For example, weak * -closures play an important role in a recent duality result in vector optimization in Grad and Jamneshan [26] which is useful for applications such as minimization of conditional risk measures. Now, by [47, Theorem X.2.9], Π 1 1 -CA 0 is required to prove that weak *closures in the normal dual of a separable Banach space even exist. Moreover, it has been argued e.g. by Kohlenbach in [41] that the second-order setting, comprehensive as it is, is not sufficient for important areas like functional analysis or topology, and should therefore be extended to higher-order frameworks that also allow for quantification over sets of real numbers, sets of sets of real numbers, sets thereof etc.
It would thus be worthwhile (1) to consider the validity of stronger comprehension axioms like Π 1 1 -CA 0 (see [47]) or, more generally, Π n 1 -CA 0 in our conditional model, and (2) to come up with conditional interpretations of higher-order systems and a corresponding transfer principle. The question whether Π 1 1 -CA 0 holds in our model is not immediately answered by the method used above for ACA 0 . For this purpose, it would be necessary to consider projections of sets of real numbers, which would require working in a conditional power set of the conditional real numbers.
Nevertheless, we are optimistic that such extensions can be constructed. Indeed, following the general construction of a conditional power set in [15], more precisely its variant defined in [36], the conditional power set of the conditional power set of the conditional natural numbers can be formed, which indicates the necessary construction needed to prove the validity of stronger comprehension schemes and a transfer principle for higher-order systems. We plan to develop this in detail in future work.