Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 651–667

On Products of Admissible Liftings and Densities

N. D. Macheras, K. Musiał and W. Strauss

Dedicated to Prof. Dr. B. Volkmann on the occasion of his 70^{th} birthday

Abstract. We introduce the classes of admissible densities and show that there exist densities respecting coordinates, and product densities with marginals prescribable within these classes. For liftings there is the corresponding class of admissibly generated liftings. We apply these results to improve theorems on product liftings and liftings respecting coordinates and to provide them a unifying approach.

Keywords: Admissible densities, product liftings, liftings respecting coordinates AMS subject classification: Primary 28 A 51, secondary 60 A 10

0. Introduction

For liftings in products of probability spaces the following problems turned up.

1^{0} The problem of the existence of liftings compatible with the product structure such as “consistent liftings” (see [12]), “product liftings and densities” (see [8 - 10]), and “liftings respecting coordinates” (see [1,2 ]), where the last one is the most farreaching concept. These liftings apply to the regularization of stochastic processes, to measurability problems for empirical processes (see [14, 15]), to the construction of strong liftings on products (see [7, 8]), and to stable measurable sets (see [13]).

As far as the existence of such liftings in full generality is unknown one asks for densities with the corresponding properties instead. For incomplete probability spaces (such as those based on Borelian and Baire-σ-algebras which occur very often in applications to probability theory) where liftings possibly will not exist (see [11]), one must resort to densities which still exist there by [3]. Our main result, Theorem 2.5, tells us that this is possible for densities respecting coordinates. For this reason our concern in this paper is with (possibly) incomplete probability spaces.

2^{0} The problem of the existence of liftings and densities listed in Problem 1^{0} with marginals prescribable to some extent. It is well known from [8] and [9] that the latter provides a tool for the construction of strong liftings in products from liftings in the factors, thus responding to a problem posed by Kupka [6].

N. D. Macheras: Dept. Stat. Univ., 80 Karaoli and Dimitriou str., 185 34 Piraeus, Greece e-mail: macheras@unipi.gr K. Musiał: Inst. Math. Univ., 50-384 Wrocław, Poland e-mail: musial@math.uni.wroc.pl W. Strauss: Math. Inst. A Univ., PF 80 11 40, D-70511 Stuttgart e-mail: strauss@mathematik.uni-stuttgart.de ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

While Problem 1^{0} naturally and historically comes first, any solution of Problem 2^{0} implies a solution of Problem 1^{0}. It has already been observed by Talagrand [15] that not all liftings have good properties from the product point of view (compare also [8]). Namely, there are complete probability spaces (Ω_{i}, Σ_{i}, μ_{i}) for i ∈ I ≠ ∅ such that for given liftings τ_{i} for μ_{i} there exists in general no lifting φ for the product measure ⊗_{i∈I}μ_{i} respecting coordinates or even being only a product lifting, having for each i ∈ I the lifting τ_{i} as its marginal.

This raises the question, whether for a given probability space there exists a class of densities whose elements can always be prescribed as marginals for densities respecting coordinates or for product densities? In this paper we describe a large class of densities of this sort, called “admissible densities”. For liftings a corresponding problem exists too and we answer it by introducing the class of “admissibly generated liftings”.

The basic Theorem 2.5 then tells us that for densities respecting coordinates there is a completely free choice of marginals in one coordinate fixed in advance, while in all other coordinates there is a free choice within the class of admissible densities. This result improves the corresponding result for product densities of [10] (and clearly implies (with different method of proof) the existence results from [2] for densities respecting coordinates).

The class of the admissibly generated liftings (see Section 3) enables us to provide a unifying approach to all known results about product liftings as well as to the partial results on liftings respecting coordinates. In particular, we present a new proof of the existence of the consistent lifting of Talagrand.

1. Preliminaries

For a given probability space (Ω, Σ, μ) a set N ∈ Σ with μ(N) = 0 is called a μ-null set, and for f, g ∈ L^{∞}(μ) and A, B ∈ Σ we write f = g a.e. (μ) or A = B a.e. (μ) if {ω ∈ Ω : f(ω) ≠ g(ω)} or A△B (the symmetric difference of A and B) is a μ-null set, respectively. By Σ_{0} will be denoted the set of all μ-null subsets of Ω. If η is a σ-subalgebra of Σ, we write E_{η}(f) for a version of the conditional expectation of f ∈ L^{∞}(μ) (the space of all bounded Σ-measurable functions on Ω) with respect to η. By (Ω, Σ̂, μ̂) there is denoted the (Carathéodory) completion of (Ω, Σ, μ).

By ℕ and ℝ we denote the sets of all natural and real numbers, respectively. If γ is an ordinal, then we will identify it with the set of all ordinals less than γ.

We use the notion of lifting and lower density (or density, for short) in the sense of Definitions 3 and 4, respectively, of [5: Chapter III, Section 1], and for each probability space (Ω, Σ, μ) we denote by Λ(μ) and ϑ(μ) the systems of all liftings and densities, respectively. For each lifting ρ on Σ̂ there exists exactly one (multiplicative) lifting ρ̃ (in the sense of [5: Chapter III, Section 1, Definition 2] on L^{∞}(μ) such that ρ̃(χ_{A}) = χ_{ρ(A)} for all A ∈ Σ (χ_{A} denotes the characteristic function of A) and vice versa (see [5: pp. 35 - 36]). For simplicity we write ρ = ρ̃ throughout.

If I is a non-empty set and ⟨(Ω_{i}, Σ_{i}, μ_{i})⟩_{i∈I} is a family of arbitrary (possibly incomplete) probability spaces, then for each ∅ ≠ J ⊆ I we denote by (Ω_{J}, Σ_{J}, μ_{J}) the uncompleted product measure space ⊗_{i∈J}(Ω_{i}, Σ_{i}, μ_{i}). It must be carefully noted that in this paper we are looking for the uncompleted product in contrast to former papers such as [2] and that the completion of (Ω_{J}, Σ_{J}, μ_{J}) = ⊗_{i∈J}(Ω_{i}, Σ_{i}, μ_{i}) will always be written (Ω_{J}, Σ̂_{J}, μ̂_{J}) = ˆ⊗_{i∈J}(Ω_{i}, Σ_{i}, μ_{i}). For any J, K ⊆ I with J ⊆ K we denote by p_{JK} the canonical projection from Ω_{K} onto Ω_{J}. For any J ⊆ I the canonical projection of Ω_{I} onto Ω_{J} is denoted by p_{J} and the σ-algebra p_{J}^{−1}(Σ_{J}) ⊆ Σ_{I} is denoted by Σ_{J}^{∗}. For every non-empty index set I, I^{∗} will denote the family of all non-empty proper subsets of I.

We say that φ ∈ ϑ(μ_{I}) satisfies condition (∗) (see [2]), if for any J, K ⊆ I with J ∩ K = ∅ we have

φ(E ∪ F) = φ(E) ∪ φ(F)  for any E ∈ Σ_{J}^{∗} and F ∈ Σ_{K}^{∗}.  (∗)

For a family ((Ω_{i}, Σ_{i}, μ_{i}))_{i∈I} of probability spaces and a probability space (Ω, Σ, μ) such that Ω = Ω_{I}, Σ ⊇ Σ_{I}, μ|Σ_{I} = μ_{I} we call a lifting π for μ a product lifting of the liftings ρ_{i} for μ_{i} (i ∈ I), and we write π ∈ ⊗_{i∈I}ρ_{i}, if the equation

π([A_{i_{1}}, ..., A_{i_{n}}]) = [ρ_{i_{1}}(A_{i_{1}}), ..., ρ_{i_{n}}(A_{i_{n}})]  (P)

holds true for all n ∈ ℕ, i_{1}, ..., i_{n} ∈ I and all A_{i_{k}} ∈ Σ_{i_{k}} (k = 1, ..., n), where [A_{i_{1}}, ..., A_{i_{n}}] denotes the cylinder set ∏_{i∈I} B_{i} for B_{i_{k}} = A_{i_{k}} (k = 1, ..., n) and B_{i} = Ω_{i} (i ∈ I \ {i_{1}, ..., i_{n}}). If I = {1, …, n}, then we write π ∈ ρ_{1} ⊗ ⋯ ⊗ ρ_{n}.

We say that φ ∈ ϑ(μ_{I}) (or π ∈ Λ(μ_{I})) respects coordinates if for each J ⊆ I the inclusion φ(Σ_{J} × Ω_{J^{c}}) ⊆ Σ_{J} × Ω_{J^{c}} holds true.

If (Ω, Σ, μ) is a probability space and I is a non-empty set, we write μ^{I} for the product measure on Ω^{I} and Σ^{I} for its domain. A lifting ρ ∈ Λ(μ̂) is consistent if for every n ∈ ℕ there exists ρ^{n} ∈ Λ( ˆ⊗^{n}μ) such that

ρ^{n}(A_{1} × ⋯ × A_{n}) = ρ(A_{1}) × ⋯ × ρ(A_{n})  (C)

for all A_{1}, …, A_{n} ∈ Σ (see Talagrand [12: Theorem 12]). We use a similar definition for densities instead of liftings.

2. Densities respecting coordinates

In the next definition we single out a class of densities with good properties from the product point of view.

Definition 2.1. Let (Θ, T, ν) be a probability space. A density υ ∈ ϑ(ν) is called admissible if it can be constructed with the help of the transfinite induction in the way described below.

(A) Let D be the smallest cardinal with the property that there exists a collection M ⊂ T such that σ(M) is dense in T in the pseudometric generated by ν. Let M = (M_{α})_{α<κ} be numbered by ordinals less than κ, where κ is the first ordinal of the cardinality D. Denote by η_{0} the σ-algebra σ(T_{0}) and for each 1 ≤ α ≤ κ denote by η_{α} the σ-algebra generated by the family {M_{γ}}_{γ<α} ∪ η_{0}. We assume that M_{α} ∉ η_{α} for each α. It is clear that without loss of generality, we may do so.

(B) υ_{0} ∈ ϑ(ν|η_{0}) is the only existing density on (Θ, η_{0}, ν|η_{0}), i.e.

υ_{0}(B) = \{ \begin{matrix}∅ & if B ∈ T_{0} \\ Θ & if B ∉ T_{0}. \end{matrix}

(C) If γ ≤ κ is a limit ordinal of uncountable cofinality, then η_{γ} = ⋃_{α<γ} η_{α} and we define υ_{γ} ∈ ϑ(ν|η_{γ}) by setting

υ_{γ}(B) := υ_{α}(B) if B ∈ η_{α} and α < γ.

(D) Assume now that there exists an increasing sequence (γ_{n}^{γ}) of ordinals that is cofinal to γ ≤ κ. For simplicity put υ_{n} := υ_{γ_{n}^{γ}} and η_{n} := η_{γ_{n}^{γ}} for all n ∈ ℕ. Then η_{γ} = σ(∪_{n∈ℕ}η_{n}) and we can define υ_{γ} by setting

υ_{γ}(B) := ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} υ_{m}(\{E_{η_{m}}(χ_{B}) > 1 − \frac{1}{k} \}) for B ∈ η_{γ}.

It follows by [3: Lemma 1] that υ_{γ} ∈ ϑ(ν|η_{γ}) and υ_{γ}|η_{n} = υ_{n} for each n ∈ ℕ.

(E) Let now γ = β + 1. To simplify the notations let M := M_{β}. It is well known that

η_{γ} = {(G ∩ M) ∪ (H ∩ M^{c}) : G, H ∈ η_{β}}.

Let M_{1} ⊇ M and M_{2} ⊇ M^{c} be η_{β}−envelopes of M and M^{c}, respectively, i.e. M_{1}, M_{2} ∈ η_{β}, (ν|η_{β})_{∗}(M_{1} \ M) = 0 and (ν|η_{β})_{∗}(M_{2} \ M^{c}) = 0 ((ν|η_{β})_{∗} is the inner measure induced by ν|η_{β}). Define

υ_{γ}((G ∩ M) ∪ (H ∩ M^{c})) :=

(M ∩ υ_{β}((G ∩ M_{1}) ∪ (H ∩ M_{1}^{c}))) ∪ (M^{c} ∩ υ_{β}((G ∩ M_{2}) ∪ (H ∩ M_{2}^{c})))

for G, H ∈ η_{β}. By [3: Lemma 2] it then follows that υ_{γ} ∈ ϑ(ν|η_{γ}) and υ_{γ}|η_{β} = υ_{β}.

(F) We define υ ∈ ϑ(ν) just by setting υ = υ_{κ}. Throughout, the collection of all admissible densities on (Θ, T, ν) will be denoted by Aϑ(ν) and each υ ∈ Aϑ(ν) will be considered together with all elements involved into the above construction without any additional remarks.

Proposition 2.2. For each probability space (Θ, T, ν) we have Aϑ(ν) ≠ ∅.

This follows by converting the above definition into an inductive proof.

Theorem 2.3. Let (Θ, T, ν) be an arbitrary probability space. If υ ∈ Aϑ(ν), then for each (Ω, Σ, μ) and each τ ∈ ϑ(μ) there exists φ ∈ ϑ(μ ⊗ ν) such that

φ(A × B) = τ(A) × υ(B) for all A ∈ Σ and B ∈ T.

If (Ω, Σ, μ) = ⊗_{i∈I}(Ω_{i}, Σ_{i}, μ_{i}) and τ ∈ ⊗_{i∈I}τ_{i} respects coordinates, then φ can be chosen to respect coordinates also. If moreover τ satisfies condition (∗), then also φ can be chosen to satisfy (∗).

Proof. Let there be given τ ∈ ϑ(μ) and υ ∈ Aϑ(ν) alltogether with other elements involved in the construction of υ ∈ Aϑ(ν). In particular, the family M = (M_{α})_{α<κ}, the σ-subalgebras (η_{α})_{α<κ}, and the sequences (γ_{n}) cofinal with limit ordinals are fixed.

Using transfinite induction, we shall be constructing now a transfinite sequence (φ̂_{α})_{α<κ} with φ̂_{α} ∈ ϑ(μ ⊗ ν|Σ ⊗ η_{α}) such that

φ̂_{α}(A × B) = τ(A) × υ_{α}(B)  for all A ∈ Σ and B ∈ η_{α}  (1)

and

φ̂_{β}|Σ ⊗ η_{α} = φ̂_{α}  for α < β < κ.  (2)

Moreover, we assume that if (Ω, Σ, μ) = ⊗_{i∈I}(Ω_{i}, Σ_{i}, μ_{i}) and τ ∈ ⊗_{i∈I}τ_{i} respects coordinates and satisfies condition (∗), then each φ̂_{α} respects the coordinates of the product space ⊗_{i∈I}(Ω_{i}, Σ_{i}, μ_{i}) ⊗ (Θ, η_{α}, ν|η_{α}) and satisfies condition (∗).

For E ∈ Σ ⊗ η_{0} we have (μ ⊗ ν)(E) = ∫ ν(E_{ω}) dμ(ω). Then Ẽ := {ω ∈ Ω : ν(E_{ω}) = 1} ∈ Σ by [4: Formulas (21.4) and (21.8)], and E = Ẽ × Θ a.e. (μ ⊗ ν). Hence if we define

φ̂_{0}(E) = τ(Ẽ) × Θ for all E = Ẽ × Θ a.e. (μ ⊗ ν),

we have φ̂_{0} ∈ ϑ(μ ⊗ ν)|Σ ⊗ η_{0}) and φ̂_{0}(A × B) = τ(A) × υ_{0}(B) for A ∈ Σ and B ∈ η_{0}. Note that φ̂_{0} respects coordinates of the space ⊗_{i∈I}(Ω_{i}, Σ_{i}, μ_{i})⊗(Θ, η_{0}, ν|η_{0}) and satisfies condition (∗).

Assume now that given γ < κ, a system (φ̂_{α}) satisfying the required conditions (1) and (2) has been constructed for all α < γ.

We have to distinguish three cases.

A) γ is a limit ordinal of uncountable cofinality: Then

Σ ⊗ η_{γ} = ⋃_{α<γ}(Σ ⊗ η_{α}).  (3)

Setting

φ̂_{γ}(E) = φ̂_{α}(E) if E ∈ Σ ⊗ η_{α},

we get unambiguously defined densities φ̂_{γ} ∈ ϑ(μ ⊗ ν|Σ ⊗ η_{γ}) such that φ̂_{γ}|Σ ⊗ η_{α} = φ̂_{α} for all α < γ. It is a direct consequence of relation (3) that condition (1) is satisfied. Clearly, φ̂_{γ} respects the coordinates of the space (⊗_{i∈I}(Ω_{i}, Σ_{i}, μ_{i})) ⊗ (Θ, η_{γ}, ν|η_{γ}) and satisfies condition (∗).

B) γ is of countable cofinality: For simplicity put υ_{n} := υ_{γ_{n}^{γ}}, φ̂_{n} := φ̂_{γ_{n}^{γ}} and η_{n} := η_{γ_{n}^{γ}} for all n ∈ ℕ. Then

Σ ⊗ η_{γ} = σ(∪_{n∈ℕ}(Σ ⊗ η_{n})).

Hence, we can define

φ̂_{γ}(P) := ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{P}) > 1 − \frac{1}{k} \}) for P ∈ Σ ⊗ η_{γ}.

It follows by [3: Lemma 1] that φ̂_{γ} ∈ ϑ(μ ⊗ ν|Σ ⊗ η_{γ}) and φ̂_{γ}|Σ ⊗ η_{n} = φ̂_{n} for each n ∈ ℕ. Now for A ∈ Σ and B ∈ η_{γ} we have

φ̂_{γ}(A × B) := ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A×B}) > 1 − \frac{1}{k} \}).

But in virtue of [8: Lemma 2.1] we have

\{E_{Σ⊗η_{m}}(χ_{A×B}) > 1 − \frac{1}{k} \} = \{E_{Σ⊗η_{m}}(χ_{A} ⊗ χ_{B}) > 1 − \frac{1}{k} \}

= \{(E_{Σ}(χ_{A}) ⊗ E_{η_{m}}(χ_{B})) > 1 − \frac{1}{k} \}

= \{(χ_{A} ⊗ E_{η_{m}}(χ_{B})) > 1 − \frac{1}{k} \}

= A × \{E_{η_{m}}(χ_{B}) > 1 − \frac{1}{k} \} a.e. (μ ⊗ ν|Σ ⊗ η_{m})

for m, k ∈ ℕ. This implies that

φ̂_{γ}(A × B) = ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} φ̂_{m}(A × \{E_{η_{m}}(χ_{B}) > 1 − \frac{1}{k} \})

= ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} (τ(A) × υ_{m}(\{E_{η_{m}}(χ_{B}) > 1 − \frac{1}{k} \})

= τ(A) × ( ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} υ_{m}(\{E_{η_{m}}(χ_{B}) > 1 − \frac{1}{k} \}))

= τ(A) × υ_{γ}(B),

i.e. φ̂_{γ}(A × B) = τ(A) × υ_{γ}(B) for all A ∈ Σ and B ∈ η_{γ}.

In order to check whether φ̂_{γ} respects coordinates, it sufficies to consider sets P ∈ Σ ⊗ η_{γ} of the form P = Q × Ω_{J^{c}} where J ∈ I^{∗}, and Q ∈ Σ_{J} ⊗ η_{γ} of the form P = Q × Ω_{J^{c}} × Θ where Q ∈ Σ_{J}. Since τ respects coordinates and φ̂_{γ} is the product of τ and υ_{γ}, we have in the second case

φ̂_{γ}(P) = φ̂_{γ}(Q × Ω_{J^{c}} × Θ) = τ(Q × Ω_{J^{c}}) × Θ = A × Ω_{J^{c}} × Θ

where A ∈ Σ_{J}, i.e. φ̂_{γ} respects the coordinates of P. In the first case, we have for each m ∈ ℕ

\begin{aligned}\{E_{Σ⊗η_{m}}(χ_{Q×Ω_{J^{c}}}) > 1 − \frac{1}{k} \} & = \{(E_{Σ_{J} ⊗η_{m}}(χ_{Q}) · (χ_{Ω_{J^{c}}})) > 1 − \frac{1}{k} \} \\ & = Ω_{J^{c}} × \{E_{Σ_{J} ⊗η_{m}}(χ_{Q}) > 1 − \frac{1}{k} \} a.e. (μ ⊗ ν|Σ ⊗ η_{m}). \end{aligned}

Since all φ̂_{m} respect coordinates, we have

\begin{aligned}φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{Q×Ω_{J^{c}}}) > 1 − \frac{1}{k} \}) & = φ̂_{m}(Ω_{J^{c}} × \{E_{Σ_{J} ⊗η_{m}}(χ_{Q}) > 1 − \frac{1}{k} \}) \\ & = Ω_{J^{c}} × A_{m} \end{aligned}

where A_{m} ∈ Σ_{J^{c}} ⊗ η_{m}. It follows that φ̂_{γ}(P) ∈ Σ_{J^{c}} ⊗ η_{γ}, i.e. φ̂_{γ} respects coordinates. To show that φ̂_{γ} satisfies condition (∗), we need the following

Claim 1. For arbitrary J, K ⊆ I_{γ} := I ∪ {γ} with J ∩ K = ∅, A ∈ Σ_{J}^{∗} ⊗ η_{γ}, B ∈ Σ_{K}^{∗} ⊗ η_{γ}, and m, k ∈ ℕ we have a.e. (μ ⊗ ν|Σ ⊗ η_{m}) the condition

\{E_{Σ⊗η_{m}}(χ_{A∪B}) > 1 − \frac{1}{k} \} = \{E_{Σ⊗η_{m}}(χ_{A}) > 1 − \frac{1}{k} \} ∪ \{E_{Σ⊗η_{m}}(χ_{B}) > 1 − \frac{1}{k} \}.

Proof. Let J, K, A, B, m, k be as in the claim. For A = ∅ or B = ∅ the claim is obvious. Suppose that A ≠ ∅ and B ≠ ∅. It is clear that a.e. (μ ⊗ ν|Σ ⊗ η_{m}) we have

\{E_{Σ⊗η_{m}}(χ_{A∪B}) > 1 − \frac{1}{k} \} ⊇ \{E_{Σ⊗η_{m}}(χ_{A}) > 1 − \frac{1}{k} \} ∪ \{E_{Σ⊗η_{m}}(χ_{B}) > 1 − \frac{1}{k} \}.

To prove the converse relation, notice that without loss of generality one may assume that γ ∈ J \ K. Then B = B_{K} × Ω_{I\K} × Θ ∈ Σ ⊗ η_{m} and so

E_{Σ⊗η_{m}}(χ_{B}) = χ_{B} a.e. (μ ⊗ ν|Σ ⊗ η_{m}).

Consequently, we get a.e. (μ ⊗ ν|Σ ⊗ η_{m}) that

\{E_{Σ⊗η_{m}}(χ_{A∪B}) > 1 − \frac{1}{k} \} = \{E_{Σ⊗η_{m}}(χ_{A}) + χ_{B} − E_{Σ⊗η_{m}}(χ_{A})χ_{B} > 1 − \frac{1}{k} \}. (4)

If (ω, θ) ∈ B, then

E_{Σ⊗η_{m}}(χ_{A})(ω, θ) + χ_{B}(ω, θ) − E_{Σ⊗η_{m}}(χ_{A})(ω, θ)χ_{B}(ω, θ) = χ_{B}(ω, θ),

hence we get a.e. (μ ⊗ ν|Σ ⊗ η_{m}) that

\begin{matrix}\{E_{Σ⊗η_{m}}(χ_{A}) + χ_{B} − E_{Σ⊗η_{m}}(χ_{A})χ_{B} > 1 − \frac{1}{k} \} ∩ B \\ = \{χ_{B} > 1 − \frac{1}{k} \} ∩ B \\ ⊆ (\{E_{Σ⊗η_{m}}(χ_{A}) > 1 − \frac{1}{k} \} ∩ B) ∪ (\{E_{Σ⊗η_{m}}(χ_{B}) > 1 − \frac{1}{k} \} ∩ B). \end{matrix} (5)

If (ω, θ) ∉ B, then

E_{Σ⊗η_{m}}(χ_{A})(ω, θ) + χ_{B}(ω, θ) − E_{Σ⊗η_{m}}(χ_{A})(ω, θ)χ_{B}(ω) = E_{Σ⊗η_{m}}(χ_{A})(ω, θ),

hence we have a.e. (μ ⊗ ν|Σ ⊗ η_{m}) that

\begin{matrix}\{E_{Σ⊗η_{m}}(χ_{A}) + χ_{B} − E_{Σ⊗η_{m}}(χ_{A})χ_{B} > 1 − \frac{1}{k} \} ∩ B^{c} \\ = \{χ_{B} > 1 − \frac{1}{k} \} ∩ B^{c} \\ ⊆ (\{E_{Σ⊗η_{m}}(χ_{A}) > 1 − \frac{1}{k} \} ∩ B^{c}) ∪ (\{E_{Σ⊗η_{m}}(χ_{B}) > 1 − \frac{1}{k} \} ∩ B^{c}). \end{matrix} (6)

From (4) - (6) it follows that a.e. (μ ⊗ ν|Σ ⊗ η_{m}) we have

\{E_{Σ⊗η_{m}}(χ_{A∪B}) > 1 − \frac{1}{k} \} ⊆ \{E_{Σ⊗η_{m}}(χ_{A}) > 1 − \frac{1}{k} \} ∪ \{E_{Σ⊗η_{m}}(χ_{B}) > 1 − \frac{1}{k} \}.

This completes the proof of Claim 1

Claim 2. If a set D ∈ Σ_{I} ⊗ η_{γ} depends on the coordinates J ∪ {γ}, then the same is true for E_{Σ_{I}⊗η_{m}}(χ_{D}).

Proof. If D = D_{J} × Ω_{J^{c}} with D_{J} ∈ Σ_{J} ⊗ η_{γ}, then
E_{Σ⊗η_{m}}(χ_{D_{J} ×ΩJ^{c}}) = E_{(Σ_{J} ⊗η_{m})×Ω_{J^{c}}} (χ_{D_{J}} ⊗ χ_{Ω_{J^{c}}}) = E_{Σ_{J} ⊗η_{m}}(χ_{D_{J}}) ⊗ χ_{Ω_{J^{c}}}

and the claim is proved

Let J, K, A, B be as in Claim 1. Assume that

A = A_{J} × Ω_{I\J}  and  B = B_{K} × Ω_{I\K} × Θ

where A_{J} ∈ Σ_{J} ⊗ η_{γ} and B_{K} ∈ Σ_{K}. Applying the inductive assumption and the above claims we have

φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A∪B}) > \frac{1}{k} \})

= φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A}) > \frac{1}{k} \} ∪ \{E_{Σ⊗η_{m}}(χ_{B}) > \frac{1}{k} \})

= φ̂_{m}(\{E_{Σ_{J} ⊗η_{m}}(χ_{A_{J}}) ⊗ χ_{Ω_{I\J}} > \frac{1}{k} \} ∪ \{E_{Σ_{K}} (χ_{B_{K}} ⊗ χ_{Ω_{I\K}} ⊗ χ_{Θ} > \frac{1}{k} \})

= φ̂_{m}(\{E_{Σ_{J} ⊗η_{m}}(χ_{A_{J}}) ⊗ χ_{Ω_{I\J}} > \frac{1}{k} \}) ∪ φ̂_{m}(\{E_{Σ_{K}} (χ_{B_{K}}) ⊗ χ_{Ω_{I\K}} ⊗ χ_{Θ} > \frac{1}{k} \})

= φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A}) > \frac{1}{k} \}) ∪ φ̂_{m}(B).

Then

φ̂_{γ}(A ∪ B) = ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A∪B}) > 1 − \frac{1}{k} \})

= ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} [φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A}) > \frac{1}{k} \}) ∪ φ̂_{m}(B)]

= ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} [φ̂_{m}(\{E_{Σ⊗η_{m}}(χ_{A}) > \frac{1}{k} \}) ∪ φ̂_{γ}(B)]

= φ̂_{γ}(A) ∪ φ̂_{γ}(B),

i.e. φ̂_{γ} satisfies condition (∗).

C) γ = β + 1: To simplify the notations let M := M_{β}. It is well known that

Σ ⊗ η_{γ} = {(K ∩ (Ω × M)) ∪ (L ∩ (Ω × M^{c})) : K, L ∈ Σ ⊗ η_{β}}.

Let M_{1} ⊇ M and M_{2} ⊇ M^{c} be η_{β}-envelopes of M and M^{c}, respectively, used in the process of describing υ_{γ}. An easy calculation shows that

E_{1} = Ω × M_{1}  and  E_{2} = Ω × M_{2}  (7)

are Σ ⊗ η_{β}-envelopes of Ω × M and Ω × M^{c}, respectively. Define

φ̂_{γ}[(K ∩ (Ω × M)) ∪ (L ∩ (Ω × M^{c}))] :=

((Ω × M) ∩ φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c}))) ∪ ((Ω × M^{c}) ∩ φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c})))

for K, L ∈ Σ ⊗ η_{β}. By [3: Lemma 2] it then follows that φ̂_{γ} ∈ ϑ(μ ⊗ ν|Σ ⊗ η_{γ}) and φ̂_{γ}|Σ ⊗ η_{β} = φ̂_{β}. For A ∈ Σ and G, H ∈ η_{β} write B = (G ∩ M) ∪ (H ∩ M^{c}). Then B ∈ η_{γ} and

\begin{aligned}A × B & = A × ((G ∩ M) ∪ (H ∩ M^{c})) \\ & = ((A × G) ∩ (Ω × M)) ∪ ((A × H) ∩ (Ω × M^{c})) \end{aligned}

together with K := A × G and L := A × H ∈ Σ ⊗ η_{β}. For simplicity put E_{0} := Ω × M. By definition we have

φ̂_{γ}(A × B) = (E_{0} ∩ φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c}))) ∪ (E_{0}^{c} ∩ φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c}))).

By an application of (7) this can be rewritten as

φ̂_{γ}(A × B) = (E_{0} ∩ φ̂_{β}(A × R)) ∪ (E_{0}^{c} ∩ φ̂_{β}(A × S))

if

R := (G ∩ M_{1}) ∪ (H ∩ M_{1}^{c})  and  S := (G ∩ M_{2}) ∪ (H ∩ M_{2}^{c}).

Since R, S ∈ η_{β}, this implies

φ̂_{γ}(A × B) = (E_{0} ∩ (τ(A) × υ_{β}(R))) ∪ (E_{0}^{c} ∩ (τ(A) × υ_{β}(S))).

By means of E_{0} = Ω × M the latter formula can be transformed into

φ̂_{γ}(A × B) = τ(A) × υ_{γ}(B) for all A ∈ Σ and B ∈ η_{γ}.

Therefore φ̂_{γ} satisfies condition (1).

In order to check whether φ̂_{γ} respects coordinates, it is sufficient to consider sets P ∈ Σ ⊗ η_{γ} of the form P = Q × Ω_{J^{c}} where J ∈ I^{∗} and Q ∈ Σ_{J} ⊗ η_{γ}, and of the form P = Q × Ω_{J^{c}} × Θ where Q ∈ Σ_{J}. Since τ respects coordinates and φ̂_{γ} is a product of τ and υ_{γ}, we have in the second case

φ̂_{γ}(P) = φ̂_{γ}(Q × Ω_{J^{c}} × Θ) = τ(Q × Ω_{J^{c}}) × Θ = A × Ω_{J^{c}} × Θ

where A ∈ Σ_{J}, i.e φ̂_{γ} respects the coordinates of P.

In the first case we take K = K′ × Ω_{J^{c}} and L = L′ × Ω_{J^{c}} with K′, L′ ∈ Σ_{J} ⊗ η_{β} such that P = (K ∩ E_{0}) ∪ (L ∩ E_{0}^{c}). Then

P = (K ∩ E_{0}) ∪ (L ∩ E_{0}^{c}) = [(K′ ∩ (Ω_{J} × M)) ∪ (L′ ∩ (Ω_{J} × M^{c}))] × Ω_{J^{c}}.

Since φ̂_{β} respects coordinates, we have

φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c}))
= φ̂_{β}[((K′ × Ω_{J^{c}}) ∩ (Ω × M_{1})) ∪ ((L′ × Ω_{J^{c}}) ∩ (Ω × M_{1}^{c}))]
= φ̂_{β}[((K′ ∩ (Ω_{J} × M_{1})) ∪ (L′ ∩ (Ω_{J} × M_{1}^{c}))) × Ω_{J^{c}}]
= A × Ω_{J^{c}}

where A ∈ Σ_{J} ⊗ η_{β}. Similarly,

φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c})) = B × Ω_{J^{c}}

with B ∈ Σ_{J} ⊗ η_{β}. Consequently,

φ̂_{γ}((K ∩ E_{0}) ∪ (L ∩ E_{0}^{c}))
= [E_{0} ∩ φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c}))] ∪ [E_{0}^{c} ∩ φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c}))]
= (E_{0} ∩ (A × Ω_{J^{c}})) ∪ (E_{0}^{c} ∩ (B × Ω_{J^{c}}))
= [((Ω_{J} × M) ∩ A) ∪ ((Ω_{J} × M^{c}) ∩ B)] × Ω_{J^{c}}.

In the case of K or L from Σ_{J}^{∗} × Θ the calculations are similar. As a consequence φ̂_{γ} respects coordinates.

To show that φ̂_{γ} satisfies condition (∗), consider J, H ⊆ I_{γ} with J ∩ H = ∅. If

A = (K ∩ E_{0}) ∪ (L ∩ E_{0}^{c}) ∈ Σ_{J}^{∗} ⊗ η_{γ} and B = (B ∩ E_{0}) ∪ (B ∩ E_{0}^{c}) ∈ Σ_{H}^{∗}

where K, L ∈ Σ_{J}^{∗} ⊗ η_{β}, then applying the inductive assumption we get

φ̂_{γ}(A ∪ B) = φ̂_{γ}((K ∩ E_{0}) ∪ (L ∩ E_{0}^{c}) ∪ (B ∩ E_{0}) ∪ (B ∩ E_{0}^{c}))
= φ̂_{γ}[((K ∪ B) ∩ E_{0}) ∪ ((L ∪ B) ∩ E_{0}^{c})]
= [E_{0} ∩ φ̂_{β}(((K ∪ B) ∩ E_{1}) ∪ ((L ∪ B) ∩ E_{1}^{c}))]
⋃ [E_{0}^{c} ∩ φ̂_{β}(((K ∪ B) ∩ E_{2}) ∪ ((L ∪ B) ∩ E_{2}^{c}))]
= [E_{0} ∩ φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c}) ∪ B)]
⋃ [E_{0}^{c} ∩ φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c}) ∪ B)]
= [E_{0} ∩ φ̂_{β}((K ∩ E_{1}) ∪ (L ∩ E_{1}^{c})) ∪ φ̂_{β}(B)]
⋃ [E_{0}^{c} ∩ φ̂_{β}((K ∩ E_{2}) ∪ (L ∩ E_{2}^{c})) ∪ φ̂_{β}(B)]
= φ̂_{γ}((K ∩ E_{0}) ∪ (L ∩ E_{0}^{c})) ∪ φ̂_{β}(B)
= φ̂_{γ}(A) ∪ φ̂_{γ}(B),

i.e. φ̂_{γ} satisfies condition (∗).

We can define now φ ∈ ϑ(μ ⊗ ν) possessing the required properties just by setting φ = φ̂_{κ}. The densities are properly defined, since each element of Σ ⊗ T is measurable with respect to some Σ ⊗ η_{α}, with α ≤ κ

Lemma 2.4. Let (Ω_{i}, Ξ_{i}, μ_{i}) (i = 1, 2, 3) be probability spaces and let f : Ω_{1} × Ω_{2} × Ω_{3} → [0, 1] be a bounded Ξ_{1} ⊗ Ω_{2} ⊗ Ξ_{3}-measurable function. Then there exists a Ξ_{1} ⊗ Ω_{2} × Ω_{3}-measurable version of E_{Ξ_{1}⊗Ξ_{2}×Ω_{3}}(f).

Proof. Let D ∈ Ξ_{1} ⊗ Ξ_{2} be an arbitrary set. By the assumption, we may assume that there is a bounded Ξ_{1} ⊗ Ξ_{3}-measurable function g satisfying everywhere the equality f(ω_{1}, ω_{2}, ω_{3}) = g(ω_{1}, ω_{3}). Let a function h be given by the equality h(ω_{1}) := ∫_{Ω_{3}} g(ω_{1}, ω_{3}) dμ_{3}(ω_{3}). Since g can be uniformly approximated by measurable simple functions, we get the Ξ_{1}-measurability of h. Then, applying the Fubini theorem (cf. [4: Theorem 21.12]), we have

∫_{D×Ω_{3}} f d(μ_{1} ⊗ μ_{2} ⊗ μ_{3}) = ∫_{D} ( ∫_{Ω_{3}} g(ω_{1}, ω_{3}) d(μ_{3}))d(μ_{1} ⊗ μ_{2})
= ∫_{D} h(ω_{1}) d(μ_{1} ⊗ μ_{2})(ω_{1}, ω_{2})
= ∫_{D×Ω_{3}} h d(μ_{1} ⊗ μ_{2} ⊗ μ_{3}).

This means that h ⊗ χ_{Ω_{2}} ⊗ χ_{Ω_{3}} is a Ξ_{1} ⊗ Ω_{2} × Ω_{3}-measurable version of E_{Ξ_{1}⊗Ξ_{2}×Ω_{3}}(f)

To some extent we can prescribe the marginals of a density respecting coordinates.

Theorem 2.5. Let ⟨(Ω_{i}, Σ_{i}, μ_{i})⟩_{i∈I} (I a non-empty index set) be a family of probability spaces with product (Ω, Σ, μ). If i_{0} ∈ I is fixed, then for each τ_{i_{0}} ∈ ϑ(μ_{i_{o}}) and for arbitrary τ_{i} ∈ Aϑ(μ_{i}) with i ∈ I \ {i_{0}} there exists φ ∈ ϑ(μ) such that φ respects coordinates, and φ ∈ ⊗_{i∈I}τ_{i}.

Proof. Let κ be the first ordinal of cardinality equal to card(I). Without loss of generality, we may assume that I = κ and i_{0} = 0. Put also (X_{γ}, T_{γ}, ν_{γ}) := ⊗_{α<γ}(Ω_{α}, Σ_{α}, μ_{α}) for 1 ≤ γ ≤ κ and let p_{αγ} be the canonical projection from X_{γ} onto X_{α} whenever α < γ.

We shall be constructing inductively densities φ_{γ} ∈ ϑ(ν_{γ}) respecting coordinates, and such that

φ_{γ} ∈ ⊗_{δ<γ}τ_{δ}  for 1 ≤ γ ≤ κ  (8)

and

φ_{γ} ∘ p_{αγ}^{−1} = p_{αγ}^{−1} ∘ φ_{α}  for 1 ≤ α < γ.  (9)

To start the induction define φ_{1} := τ_{0}. Suppose that for some γ ≤ κ and all α < γ the densities φ_{α} ∈ ϑ(ν_{α}) respecting coordinates and satisfying (8) and (9) are already known. We have to distinguish three cases.

A) γ = α + 1: By Theorem 2.3 there exists a density φ_{α+1} ∈ ϑ(ν_{α+1}) respecting coordinates and satisfying the condition φ_{γ} ∈ ⊗_{β<γ}τ_{β}. Clearly, condition (9) is also satisfied.

B) γ is of countable cofinality: We assume here that if H ⊂ γ, then H^{c} := γ \ H. For each α with 1 ≤ α < γ we put T_{α}^{∗} := p_{αγ}^{−1}(T_{α}) and ν_{α}^{∗} := ν_{γ}|T_{α}^{∗}. Clearly, for 1 ≤ α ≤ β < γ

T_{α}^{∗} ⊆ T_{β}^{∗}  and  ν_{β}^{∗}|T_{α}^{∗} = ν_{α}^{∗}.

For each α < γ define a density φ_{α}^{∗} ∈ ϑ(ν_{α}^{∗}) by means of

φ_{α}^{∗}(A^{∗}) := p_{αγ}^{−1}(φ_{α}(A))

where A ∈ T_{α} and A^{∗} = p_{αγ}^{−1}(A) a.e. (ν_{α}^{∗}). It is easily seen that φ_{β}^{∗}|T_{α}^{∗} = φ_{α}^{∗} for all α and β with 1 ≤ α ≤ β < γ.

Let (γ_{n}) be an increasing sequence of ordinals cofinal with γ. Then for each α < γ there exists n ∈ ℕ such that T_{α}^{∗} ⊆ T_{n}^{∗} and φ_{n}^{∗}|T_{α}^{∗} = φ_{α}^{∗} where T_{n}^{∗} := T_{γ_{n}}^{∗} and φ_{n}^{∗} := φ_{γ_{n}}^{∗}. Hence

T_{γ} = σ( ⋃_{n∈ℕ} T_{n}^{∗})  and  φ_{n+1}^{∗}|T_{n}^{∗} = φ_{n}^{∗}

for each n ∈ ℕ. Thus, we can define

φ_{γ}(P) := ⋂_{k∈ℕ} ⋃_{n∈ℕ} ⋂_{m≥n} φ_{m}^{∗}(\{E_{T_{m}^{∗}}(χ_{P}) > \frac{1}{k} \}) for each P ∈ T_{γ}.

It follows by [3: Lemma 1] that φ_{γ} ∈ ϑ(ν_{γ}) and φ_{γ}|T_{n}^{∗} = φ_{n}^{∗} for all n ∈ ℕ, hence φ_{γ}|T_{α}^{∗} = φ_{α}^{∗} for all α < γ.

We have to show yet that φ_{γ} respects coordinates. To do it, take a non-empty set J ⊂ γ and assume that P = Q × Ω_{J^{c}} where Q ∈ Σ_{J}. Notice that for each m ∈ ℕ we have the equality

γ = (γ_{m} ∩ J) ∪ (γ_{m} ∩ J^{c}) ∪ γ_{m}^{c}.

Let Ξ_{1} := Σ_{γ_{m}∩J}, Ξ_{2} := Σ_{γ_{m}∩J^{c}} and Ξ_{3} := Σ_{γ_{m}^{c}∩γ}. Applying Lemma 2.4 to f = χ_{P}, we see that the function E_{T_{m}^{∗}}(χ_{P}) can be assumed to be Ξ_{1}⊗Ω_{γ_{m}^{c}∪J^{c}}-measurable. Since φ_{m} respects coordinates, we get a similar measurability of the set φ_{m}^{∗}(\{E_{T_{m}^{∗}}(χ_{P}) > 1− \frac{1}{k} \}). In particular, the set φ_{m}^{∗}(\{E_{T_{m}^{∗}}(χ_{P}) > 1 − \frac{1}{k} \}) is Σ_{J} × Ω_{J^{c}}-measurable. Consequently, the set φ_{γ}(P) is Σ_{J} × Ω_{J^{c}}-measurable. This proves that φ_{γ} respects coordinates.

C) γ is of uncountable cofinality: In this case T_{γ} = ⋃_{1≤α<γ} T_{α}^{∗}. Now define for each 1 ≤ α < γ a density φ_{α}^{∗} ∈ ϑ(ν_{α}^{∗}), where ν_{α}^{∗} := ν_{γ}|T_{α}^{∗}, by

φ_{α}^{∗}(A^{∗}) := p_{αγ}^{−1}(φ_{α}(A))

for each A ∈ T_{α} with A^{∗} = p_{αγ}^{−1}(A) a.e. (ν_{α}^{∗}). Since φ_{β}^{∗}|T_{α}^{∗} = φ_{α}^{∗} for all α and β with 1 ≤ α ≤ β < γ there exists a density φ_{γ} ∈ ϑ(ν_{γ}) defined for each A ∈ T_{α}^{∗} by

φ_{γ}(A) = φ_{α}^{∗}(A).

Clearly, φ_{γ} respects coordinates and φ_{γ}|T_{α}^{∗} = φ_{α}^{∗} for arbitrary 1 ≤ α < γ. Hence

φ_{γ} ∘ p_{αγ}^{−1} = p_{αγ}^{−1} ∘ φ_{α} for arbitrary 1 ≤ α < γ.

It follows from steps (B) and (C) that for each limit ordinal 1 < γ ≤ κ there exists always a density φ_{γ} ∈ ϑ(ν_{γ}) satisfying condition (9). As an immediate consequence of (9) we have

φ_{γ}(A × ∏_{α≤β<γ} Ω_{β}) = φ_{α}(A) × ∏_{α≤β<γ} Ω_{β}  (10)

for all 1 ≤ α ≤ γ ≤ κ and A ∈ T_{α}. The condition φ_{γ} ∈ ⊗_{β<γ}τ_{β} is a direct consequence of (10) and of the inductive assumption about each φ_{α} with α < γ. We can define now φ ∈ ϑ(μ_{I}) posessing the required properties just by setting φ := φ_{κ}

The following corollary, containing the main result of [10] as well as Theorem 3 of [8: Section 2] is an immediate consequence of Theorem 2.5.

Corollary 2.6. Let ⟨(Ω_{i}, Σ_{i}, μ_{i})⟩_{i∈I} be a family of probability spaces. Fix i_{0} ∈ I. Then for each τ_{i_{0}} ∈ ϑ(μ_{i_{o}}) there exist τ_{i} ∈ ϑ(μ_{i}) for i ∈ I \{i_{0}} and φ ∈ ϑ(μ_{I}) respecting coordinates and satisfying the condition φ ∈ ⊗_{i∈I}τ_{i}.

Corollary 2.7. Let (Ω, Σ, μ) be a probability space. For a non-empty set I write p_{Ii}(ω) := ω_{i} for ω ∈ Ω^{I} and i ∈ I. Then for each τ ∈ Aϑ(μ) and for each non-empty set I there exists τ_{I} ∈ ϑ(μ_{I}) such that

τ_{I}(p_{Ii}^{−1}(A)) = p_{Ii}^{−1}(τ(A))

for each A ∈ Σ and i ∈ I. In particular, each admissible density is a consistent density.

The following result is a particular case of Fremlin’s Theorem 346G in [2], proved in a different way. But here the completeness assumption for the probability spaces, used in Fremlin’s result, is avoided.

Corollary 2.8. Let ⟨(Ω_{i}, Σ_{i}, μ_{i})⟩_{i∈I} be a family of probability spaces. Then there exists φ ∈ ϑ(μ_{I}) respecting coordinates.

3. Admissibly generated liftings

The next definition is a counterpart of Definition 2.1 above and singles out a class of liftings with good properties from the product point of view.

Definition 3.1. Let υ ∈ ϑ(ν) be an arbitrary density on (Θ, T, ν). It is well known by [16] that

F(θ) := {B ∈ T : θ ∈ υ(B)}

is a filterbasis on Θ so that for each θ ∈ Θ one can choose an ultrafilter U(θ) on Θ finer than F(θ). We define then

σ(B) := {θ ∈ Θ : B ∈ U(θ)} for all B ∈ T.

It has been proven in [16] that σ ∈ Λ(ν̂) and

υ(B) ⊆ σ(B) ⊆ [υ(B^{c})]^{c}  for all B ∈ T.  (11)

An arbitrary lifting σ constructed in the above manner from a density υ will be called a lifting generated by υ. If the lifting σ described above is generated by an admissible density υ, then it is called admissibly generated (by υ) and the family of all admissibly generated liftings on (Θ, T, ν) is denoted by AGΛ(ν).

Remark. It follows from Proposition 2.2 and Definition 3.1 that AGΛ(ν) ≠ ∅.

Theorem 3.2. Let (Ω, Σ, μ) and (Θ, T, ν) be complete probability spaces and let τ ∈ ϑ(μ), υ ∈ ϑ(ν), and φ ∈ ϑ(μ ⊗ ν) be densities such that the condition φ ∈ τ ⊗ υ holds true and φ satisfies condition (∗). Then, for each ρ ∈ Λ(μ) and each σ ∈ Λ(ν) generated by τ and υ, respectively, there exists π ∈ Λ(μ ˆ⊗ν) such that

π(A × B) = ρ(A) × σ(B)  for all A ∈ Σ and B ∈ T  (P)
φ(E) ⊆ π(E) ⊆ [φ(E^{c})]^{c}  for all E ∈ Σ ⊗ T
π(f ⊗ g) = ρ(f) ⊗ σ(g)  for all f ∈ L^{∞}(μ) and g ∈ L^{∞}(ν).

Proof. For each ω ∈ Ω and each θ ∈ Θ let

F(ω) := {A ∈ Σ : ω ∈ τ(A)}  and  F(θ) := {B ∈ T : θ ∈ υ(B)}

be filterbases generated by τ and υ, respectively. Then let

U(ω) := {A ∈ Σ : ω ∈ ρ(A)}  and  U(θ) := {B ∈ T : θ ∈ σ(B)}

be the ultrafilters generated by ρ and σ, respectively, so that

F(ω) ⊆ U(ω) ⊂ Σ  and  F(θ) ⊆ U(θ) ⊂ T.

For each (ω, θ) ∈ Ω × Θ define a filterbase by

F(ω, θ) := {E ∈ Σ ⊗ T : (ω, θ) ∈ φ(E)}.

Claim 1. For each (ω, θ) ∈ Ω × Θ, A ∈ U(ω), B ∈ U(θ), and E ∈ F(ω, θ) we have

E ∩ (A × B) ≠ ∅.

Proof. Let (ω, θ), E and A, B be as in Claim 1. Assume that E ∩ (A × B) = ∅. Then we get by using condition (∗)

φ(E) ⊆ φ([A^{c} × Θ] ∪ [Ω × B^{c}])
= φ(A^{c} × Θ) ∪ φ(Ω × B^{c})
= [υ(A^{c}) × Θ] ∪ [Ω × τ(B^{c})]
⊆ [ρ(A^{c}) × Θ] ∪ [Ω × σ(B^{c})]
= [ρ(A) × σ(B)]^{c},

i.e. φ(E) ∩ [ρ(A) × σ(B)] = ∅, which contradicts to the assumption (ω, θ) ∈ φ(E) ∩ [ρ(A) × σ(B)] of Claim 1

By the above claim there exists an ultrafilter U(ω, θ) ⊆ Σ ˆ⊗T finer than F(ω, θ) such that

A × B ∈ U(ω, θ)  for all A ∈ U(ω) and B ∈ U(θ).  (12)

For each E ∈ Σ ˆ⊗T put

π(E) := {(ω, θ) ∈ Ω × Θ : E ∈ U(ω, θ)}.

It follows by [16], for example, that π ∈ Λ(μ ˆ⊗ν).

Claim 2. For each A ∈ Σ and B ∈ T we have

ρ(A) × σ(B) = π(A × B).

Proof. For A ∈ Σ and B ∈ T we get from (12) that

(ω, θ) ∈ ρ(A) × σ(B) ⟺ ω ∈ ρ(A) and θ ∈ σ(B)
⟺ A ∈ U(ω)&B ∈ U(θ)
⟹ A × B ∈ U(ω, θ)
⟺ (ω, θ) ∈ π(A × B),

i.e. ρ(A) × σ(B) ⊆ π(A × B). It remains to show that π(A × B) ⊆ ρ(A) × σ(B). Applying the first part of the proof we get for each A ∈ Σ and B ∈ T that

(ρ(A) × σ(B))^{c} = [ρ(A^{c}) × Θ] ∪ [Ω × σ(B^{c})]
⊆ π(A^{c} × Θ) ∪ π(Ω × B^{c})
= [π(A × B)]^{c}.

This completes the proof of Theorem 3.2

Corollary 3.3. Let (Θ, T, ν) be an arbitrary complete probability space. If σ ∈ AGΛ(ν), then for each complete probability space (Ω, Σ, μ) and for each ρ ∈ Λ(μ) there exists π ∈ Λ(μ ˆ⊗ν) such that π ∈ ρ ⊗ σ.

Proof. Let (Ω, Σ, μ) and ρ be as as in the corollary. Since AGΛ(ν) ≠ ∅, it follows by Theorem 2.3 that there exists φ ∈ ϑ(μ ⊗ ν) satisfying condition (∗) and such that φ ∈ ρ ⊗ τ. The result now follows from Theorem 3.2

The following corollary is the main result of [8].

Corollary 3.4 (see [8: Section 2, Theorem 4]). If (Ω, Σ, μ) and (Θ, T, ν) are complete probability spaces, then for each ρ ∈ Λ(μ) there exist σ ∈ Λ(ν) and π ∈ Λ(μ ˆ⊗ν) such that

π(A × B) = ρ(A) × σ(B)  for all A ∈ Σ and B ∈ T  (P)
π(f ⊗ g) = ρ(f) ⊗ σ(g)  for all f ∈ L^{∞}(μ) and g ∈ L^{∞}(ν).

Theorem 3.5. Let κ be an ordinal and ⟨(Ω_{α}, Σ_{α}, μ_{α})⟩_{α<κ} be a family of complete probability spaces. Moreover, for each 1 ≤ λ ≤ κ put (X_{λ}, T_{λ}, ν_{λ}) := ˆ⊗_{α<λ}(Ω_{α}, Σ_{α}, μ_{α}). Then for each ρ_{0} ∈ Λ(μ_{0}) and each collection {ρ_{α} ∈ AGΛ(μ_{α}) : 0 < α < κ} there exists a family {π_{λ} ∈ Λ(ν_{λ}) : 1 ≤ λ ≤ κ} such that:

(i) π_{λ} ∈ ⊗_{α<λ}ρ_{α} for each 1 ≤ λ ≤ κ.

(ii) π_{κ}(A × ∏_{λ≤γ<κ} Ω_{γ}) = π_{λ}(A) × ∏_{λ≤γ<κ} Ω_{γ} for each 1 ≤ λ < κ and A ∈ T_{λ}.

Proof. For each 1 ≤ λ ≤ κ let p_{αλ} be the canonical projection from X_{λ} onto X_{α} whenever α < λ. We are going to construct inductively liftings π_{λ} ∈ Λ(ν_{λ}) such that

π_{λ} ∈ ⊗_{δ<λ}ρ_{δ}  for 0 < λ ≤ κ  (13)

and

π_{λ} ∘ p_{αλ}^{−1} = p_{αλ}^{−1} ∘ π_{α}  for 0 ≤ α < λ.  (14)

To start the induction let π_{1} := ρ_{0} ∈ Λ(ν_{1}). Then assume that for 1 ≤ λ < κ liftings π_{α} ∈ Λ(ν_{α}) satisfying conditions (13) and (14) are already known. We have to distinguish three cases.

A) λ = α + 1: By Corollary 3.3 there exists a lifting π_{α+1} ∈ Λ(ν_{α+1}) satisfying condition (13). Clearly, condition (14) is also satisfied by π_{α+1}.

B) λ is of countable cofinality: For each α with 0 ≤ α < λ denote by T_{α}^{∗} the σ-algebra T_{α}^{∗} ∪ (T_{λ})_{0}, and ν_{α}^{∗} := ν_{λ}|T_{α}^{∗}. Clearly, for 0 ≤ α ≤ β < λ, T_{α}^{∗} ⊆ T_{β}^{∗} and ν_{β}^{∗}(A) = ν_{α}^{∗}(A) holds for any A ∈ T_{α}^{∗}. For each α < λ define a lifting π_{α}^{∗} ∈ Λ(ν_{α}^{∗}) by

π_{α}^{∗}(A^{∗}) := p_{αλ}^{−1}(π_{α}(A))

where A^{∗} ∈ T_{α}^{∗} and A ∈ T_{α} with A^{∗} = p_{αλ}^{−1}(A) a.e. (ν_{α}^{∗}). It is easily seen that, for all α and β with 0 ≤ α ≤ β < λ, π_{β}^{∗}|T_{α}^{∗} = π_{α}^{∗} holds. But since λ is of countable cofinality for each α < λ there exists n ∈ ℕ such that T_{α}^{∗} ⊆ T_{n}^{∗} := T_{α_{n}}^{∗} and π_{n}^{∗}|T_{α}^{∗} = π_{α}^{∗} where π_{n}^{∗} := π_{α_{n}}^{∗} and (α_{n}) is an increasing sequence cofinal with λ. Hence π_{n+1}|T_{n}^{∗} = π_{n} for each n ∈ ℕ.

It follows easily from [5: Theorem IV.2] that there exists π_{λ} ∈ Λ(ν_{λ}) being the common extension of all π_{α}^{∗} with α < λ. Consequently, the family (π_{α})_{α≤λ} satisfies condition (14). The relation π_{λ} ∈ ⊗_{δ<λ}ρ_{δ} is an immediate consequence of the inductive assumption about π_{α} with α < λ, since λ is a limit ordinal.

C) The limit ordinal λ is of uncountable cofinality: With the notation from the Case B) we have now T_{λ} = ⋃_{0≤α<λ}(T_{α}^{∗}). Since, for all α and β with 0 ≤ α ≤ β < λ, π_{β}^{∗}|T_{α}^{∗} := π_{α}^{∗} holds, there exists a lifting π_{λ} ∈ Λ(ν_{λ}) such that π_{λ}|T_{α}^{∗} = π_{α}^{∗} for each 1 < λ ≤ κ. Consequently, the family (π_{α})_{α≤λ} satisfies condition (14). The relation π_{λ} ∈ ⊗_{δ<λ}ρ_{δ} follows in the same way as before

The following corollary is the main result of [9].

Corollary 3.6 (see [9: Section 2, Theorem 1]). Let be given a family ((Ω_{i}, Σ_{i}, μ_{i}))_{i∈I} (I a non-empty index set) of complete probability spaces. Fix i_{0} ∈ I. Then for each ρ_{i_{0}} ∈ Λ(μ_{i_{o}}) there exist ρ_{i} ∈ Λ(μ_{i}) for i ∈ I \ {i_{0}} and π ∈ Λ(μ̂_{I}) such that π ∈ ⊗_{i∈I}ρ_{i}.

Proof. Let κ be the first ordinal of cardinality equal to card(I). Without loss of generality we may assume that I = κ and i_{0} = 0. Since AGΛ(μ_{i}) ≠ ∅ for i ∈ I \ {i_{0}} the result follows immediately by Theorem 3.5

Corollary 3.7 (see [2: Theorem 346H]). Let κ be an ordinal and ⟨(Ω_{α}, Σ_{α}, μ_{α})⟩_{α<κ} be any family of complete probability spaces with completed product (Ω, Σ, μ). Then there exists a π ∈ Λ(μ) such that π(A) ∈ Σ_{J}^{∗} whenever A ∈ Σ_{J}^{∗} and J ⊆ κ is either a singleton or an initial segment of κ.

The following corollary is a generalization of [2: Theorem 346I].

On Products of Admissible Liftings 667

Corollary 3.8. Let (Ω, Σ, μ) be a complete probability space. For any non-empty set I write p_{Ii}(ω) := ω_{i} for ω ∈ Ω^{I} and i ∈ I. Then for each ρ ∈ AGΛ(μ) and for each non-empty set I there exists ρ_{I} ∈ Λ(μ^{I}) such that

ρ_{I}(p_{Ii}^{−1}(A)) = p_{Ii}^{−1}(ρ(A))

for each A ∈ Σ and i ∈ I. In particular, each admissibly generated lifting is a consistent lifting.

Corollary 3.9 (see [12: Theorem 12]). Let (Ω, Σ, μ) be a complete probability space. Then there exists a consistent lifting ρ ∈ Λ(μ).

References

[1] Burke, M. R.: Consistent liftings. Unpublished notes of 1995/01/23.

[2] Fremlin, D. H.: Measure Theory (to appear).

[3] Graf, S. and H. von Weizsäcker: On the existence of lower densities in noncomplete measure spaces. Lect. Notes Math. 541 (1976), 155 – 158.

[4] Hewitt, E. and K. Stromberg: Real and Abstract Analysis. Berlin - Heidelberg - New York: Springer-Verlag 1965.

[5] Ionescu Tulcea, A. and C.: Topics in the Theory of Lifting. Berlin - Heidelberg - New York: Springer-Verlag 1969.

[6] Kupka, J.: Strong liftings with applications to measurable cross sections in locally compact groups. Israel J. Math. Soc. 44 (1983), 243 – 261.

[7] Macheras, N. D. and W. Strauss: On strong liftings for projective limits. Fund. Math. 144 (1994), 209 – 229.

[8] Macheras, N. D. and W. Strauss: On products of almost strong liftings. J. Austral. Math. Soc. (Series A) 60 (1996), 311 – 333.

[9] Macheras, N. D. and W. Strauss: The product lifting for arbitrary products of complete probability spaces. Atti Sem. Mat. Fis. Univ. Modena 44 (1996), 485 – 496.

[10] Macheras, N. D. and W. Strauss: Products of lower densities. Z. Anal. Anw. 14 (1995), 25 – 32.

[11] Shelah, S.: Lifting problem of the measure algebra. Israel J. Math. 45 (1983), 90 – 96.

[12] Talagrand, M.: Closed convex hull of set of measurable functions and measurability of translations. Ann. Inst. Fourier (Grenoble) 32 (1982), 39 – 69.

[13] Talagrand, M.: Pettis Integral and Measure Theory (Mem. Amer. Math. Soc.: Vol. 51 (307)). Providence, Rhode Island: Amer. Math. Soc. 1984.

[14] Talagrand, M.: Measurability problems for empirical processes. Ann. Probability 15 (1987), 204 – 212.

[15] Talagrand, M.: On liftings and regularization of stochastic processes. Probab. Theor. Relat. Fields 78 (1988), 127 – 134.

[16] Traynor, T.: An elementary proof of the lifting theorem. Pac. J. Math. 53 (1974), 267 – 272.

Received 18.07.1998; in revised form 03.03.1999