ON VARIOUS MODES OF SCALAR CONVERGENCE IN L_{0}(𝔛)

S. J. Dilworth^{†} and Maria Girardi ^{†}*

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. dilworth@math.sc.edu and girardi@math.sc.edu

August 3, 2000

J. Math. Anal. Appl. 259 (2001), no. 2, 660–684

Abstract. A sequence {f_{n}} of strongly-measurable functions taking values in a Banach space 𝔛 is scalarly null a.e. (resp. scalarly null in measure) if x^{∗}f_{n} → 0 a.e. (resp. x^{∗}f_{n} → 0 in measure) for every x^{∗} ∈ 𝔛^{∗}. Let 1 ≤ p ≤ ∞. The main questions addressed in this paper are whether an L_{p}(𝔛)-bounded sequence that is scalarly null a.e. will converge weakly a.e. (or have a subsequence which converges weakly a.e.), and whether an L_{p}(𝔛)-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null a.e. The answers to these and other similar questions often depend upon p and upon the geometry of 𝔛.

1. INTRODUCTION

Consider the space L_{0}(𝔛) of strongly-measurable functions defined on the usual Lebesgue measure space (Ω, Σ, μ) on [0, 1] and taking values in the Banach space 𝔛. Among the most important linear subspaces of L_{0}(𝔛) are the Bochner-Lebesgue spaces L_{p}(𝔛) for 1 ≤ p ≤ ∞. When 𝔛 = ℝ, the L_{p}(𝔛) spaces are just the usual Lebesgue spaces, which we shall denote by L_{p}. A sequence {f_{n}} in L_{0}(𝔛) may converge to f in L_{0}(𝔛) in a variety of modes. In this paper, we examine the implications going between the four modes described below.

The sequence {f_{n}} converges scalarly a.e. (resp. scalarly in L_{p} for a fixed 1 ≤ p ≤ ∞, resp. scalarly in measure) to f if for each x^{∗} in the dual space 𝔛^{∗} of 𝔛,

†Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Participant, Convex Geometry and Geometric Functional Analysis Program at the Mathematical Sciences Research Institute, Berkeley, CA (supported in part by NSF grant DMS-9022140). *Supported in part by NSF grant DMS-9306460.

Typeset by AMS-TEX

the corresponding sequence {x^{∗}f_{n}} in L_{0}(ℝ) converges almost everywhere (resp. in L_{p}(ℝ), resp. in measure) to x^{∗}f. Thus, {f_{n}} converges scalarly a.e. to f if for each x^{∗} in 𝔛^{∗} there is a set A (which depends on x^{∗}) of full measure such that {x^{∗}f_{n}(ω)} converges to x^{∗}f(ω) for each ω in A. If the sequence satisfies the stronger property that there is a set A (independent of x^{∗}) of full measure such that {x^{∗}f_{n}(ω)} converges to x^{∗}f(ω) for each ω in A and each x^{∗} in 𝔛^{∗}, then we say that {f_{n}} converges weakly almost everywhere (i.e. weakly a.e.) to f.

The following obvious positive and negative implications hold:

weakly a.e.  ⟶  scalarly a.e.
↑∤ ↓∤  ↗ ∣ ↙ ∣ ↘ ↖ ∣  ↑∤ ↓
scalarly in L_{p}  \overrightarrow{⟵ ∣}  scalarly in measure

for each 1 ≤ p < ∞. Similarly:

weakly a.e.  ⟶  scalarly a.e.
↓∤  ↗ ↙ ∣ ↘ ↖ ∣  ↑∤ ↓
scalarly in L_{p}  \overrightarrow{⟵ ∣}  scalarly in measure

for p = ∞.

Sets of the form

V_{x^{∗},ε}(0) = {g ∈ L_{0}(𝔛): μ {|x^{∗}(g)| ≥ ε} < ε} ,

where x^{∗} ∈ 𝔛^{∗} and ε > 0, form a local subbasis at zero for the translation-invariant topology of scalar convergence in measure. Endowed with this topology, L_{0}(𝔛) is a non-locally convex Hausdorff topological vector space. For a fixed 1 ≤ p ≤ ∞, sets of the form

S_{x^{∗},ε}(0) = {g ∈ L_{p}(𝔛): ∥x^{∗}g∥_{L_{p}} < ε} ,

where x^{∗} ∈ 𝔛^{∗} and ε > 0, form a local subbasis at zero for the translation-invariant topology of scalar convergence in L_{p}. Endowed with this topology, L_{p}(𝔛) is a locally convex Hausdorff topological vector space.

Let 𝔛_{0} be a norm-closed subspace of 𝔛. The Hahn-Banach Theorem quickly gives two observations. First, a sequence of L_{0}(𝔛_{0}) functions converges to the null function in one of the above modes when viewed as a sequence in L_{0}(𝔛_{0}) if and only if it does so when viewed as a sequence in L_{0}(𝔛). Secondly, the topology of scalar convergence in measure on L_{0}(𝔛_{0}) coincides with the subspace topology inherited from the topology of scalar convergence in measure on L_{0}(𝔛). Let us show that under this identification L_{0}(𝔛_{0}) is in fact a closed subspace of L_{0}(𝔛).

Proposition 1.1. Let 𝔛_{0} be a (norm-closed) subspace of 𝔛. Then L_{0}(𝔛_{0}) is a closed subspace of L_{0}(𝔛) in the topology of scalar convergence in measure. In particular, if {f_{n}} is a sequence of 𝔛_{0}-valued functions in L_{0}(𝔛) that converges scalarly in measure to f in L_{0}(𝔛), then f is also 𝔛_{0}-valued.

Proof. Let f ∈ L_{0}(𝔛) belong to the closure of L_{0}(𝔛_{0}) in the topology of scalar convergence in measure. Since the range of f is essentially separably-valued, there is a subset Y ⊃ 𝔛_{0} such that Y/𝔛_{0} is separable and f(ω) ∈ Y a.e. By the Hahn-Banach theorem there exists a sequence {x_{n}^{∗}} in 𝔛^{∗} such that if y ∈ Y then y ∈ 𝔛_{0} if and only if x_{n}^{∗}(y) = 0 for all n ≥ 1. In particular, if g ∈ L_{0}(𝔛_{0}) then x_{n}^{∗}(g(ω)) = 0 for all n ≥ 1 a.e., and it follows that x_{n}^{∗}(f(ω)) = 0 for all n ≥ 1 a.e., which proves that f ∈ L_{0}(𝔛_{0}).

For 1 ≤ p ≤ ∞, it is easy to see that L_{p}(𝔛) is not a closed subspace of L_{0}(𝔛) in the topology of scalar convergence in measure for any Banach space 𝔛. So we consider the unit ball

B(L_{p}(𝔛)) = {g ∈ L_{0}(𝔛): ∥g∥_{L_{p}(𝔛)} ≤ 1}

of L_{p}(𝔛).

Proposition 1.2. For 1 ≤ p ≤ ∞, the unit ball B(L_{p}(𝔛)) of L_{p}(𝔛) is closed in L_{0}(𝔛) in the topology of scalar convergence in measure.

Proof. Consider f ∈ L_{0}(𝔛)\B(L_{p}(𝔛)). It is sufficient to find an open neighborhood about f that does not meet B(L_{p}(𝔛)).

Fix ε > 0 so that (1 − 4ε) ∥f∥_{L_{p}(𝔛)} > 1. Since f is strongly-measurable, there is a countably-valued function g ∈ L_{0}(𝔛) satisfying

∥f(ω) − g(ω)∥_{𝔛} ≤ ε max{∥f(ω)∥_{𝔛} , ∥g(ω)∥_{𝔛}}

for almost all ω. By making an appropriate choice of representative we may write g = ∑_{k} x_{k}1_{E_{k}}, where {E_{k}}_{k} partitions the support of g into sets of strictly positive measure. Now ∥g∥_{L_{p}(𝔛)} ≥ (1 − ε) ∥f∥_{L_{p}(𝔛)}, and so (1 − 3ε) ∥g∥_{L_{p}(𝔛)} > 1. Hence we may choose N ∈ ℕ so that

(1 − 3ε) \left∥{ \atop g̃}\right∥_{L_{p}(𝔛)} > 1 ,

where g̃ = ∑_{k=1}^{N} x_{k}1_{E_{k}}. Now find {x_{k}^{∗}}_{k=1}^{N} in S(𝔛^{∗}) with x_{k}^{∗}(x_{k}) = ∥x_{k}∥.

Consider the following neighborhood of f in the topology of scalar convergence in measure:

U = ⋂_{k=1}^{N} {h ∈ L_{0}(𝔛): μ {|x_{k}^{∗}(h − f)| ≥ ε ∥x_{k}∥} < εμ(E_{k})} .

For 1 ≤ k ≤ N, if ω ∈ E_{k} then ∥f(ω) − g̃(ω)∥ ≤ ε∥x_{k}∥; thus, for each h ∈ U

μ{ω ∈ E_{k} : ∥h(ω)∥ ≥ (1 − 2ε)∥x_{k}∥} ≥ (1 − ε)μ(E_{k}) .

Hence

∥h∥_{L_{p}(𝔛)} ≥ (1 − ε)^{1/p}(1 − 2ε) \left∥{ \atop g̃}\right∥_{L_{p}(𝔛)} > (1 − 3ε) \left∥{ \atop g̃}\right∥_{L_{p}(𝔛)} > 1 ,

following the convention that 1/∞ = 0. So U does not intersect B(L_{p}(𝔛)), as required.

This suggests imposing the following natural boundedness conditions: a sequence {f_{n}} of L_{p}(𝔛) functions is said to be:

- pointwise bounded a.e. if sup_{n} ∥f_{n}(ω)∥_{𝔛} < ∞ for each ω in some set of full measure.

- bounded in L_{p}(𝔛) (for short, L_{p}(𝔛)-bounded) if sup_{n} ∥f_{n}∥_{L_{p}(𝔛)} < ∞. From Proposition 1.2, we see that sequences of the latter type are well-behaved in the following sense.

Corollary 1.3. Let 1 ≤ p ≤ ∞ and suppose that {f_{n}} is bounded in L_{p}(𝔛). If {f_{n}} converges scalarly in measure to f ∈ L_{0}(𝔛), then f ∈ L_{p}(𝔛).

Remarks. 1. Hence, in discussing the convergence (in any one of the above four modes) of a sequence in L_{0}(𝔛) of functions valued in a subspace 𝔛_{0} of 𝔛, there is no loss of generality in taking the limit function to be the null function and viewing the sequence as in L_{0}(𝔛_{0}).

2. Similarly, if we choose to restrict ourselves to the subset L_{p}(𝔛) of L_{0}(𝔛), in discussing scalar convergence in measure for an L_{p}(𝔛)-bounded sequence, there will be no loss of generality in taking the limit function to be the null function.

3. The question of the existence of a limit for an L_{p}(𝔛)-bounded Cauchy sequence in the topology of scalar convergence of measure is more problematic and will be deferred until Section 5.

We will also use the following elementary facts without further comment. Fact 1.4 provides a useful necessary condition for weak a.e. convergence while Fact 1.5 will be used to prove scalar convergence a.e.

Fact 1.4. A weakly convergent sequence in a Banach space is norm-bounded. Thus, if for a given sequence {f_{n}} in L_{0}(𝔛), there exists a subset B of strictly positive μ-measure such that lim sup ∥f_{n}(ω)∥ = ∞ for each ω ∈ B, then {f_{n}} does not converge weakly a.e.

Fact 1.5. A sequence {f_{n}} in L_{1} converges to the null function a.e. whenever ∑ ∥f_{n}∥_{L_{1}} < ∞.

If Y is a subset of 𝔛, then sp Y denotes the linear span of Y and [Y] denotes the closed linear span of Y. All notation and terminology, not otherwise explained, are as in [3] or [14].

2. CONVERGENCE PROPERTIES

Proposition 1.2 suggests that it would be of interest to study the following properties that a Banach space 𝔛 might enjoy.

(A′_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly a.e. also

converges weakly a.e. (A_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly a.e. has

a subsequence that converges weakly a.e. (B_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly in mea-

sure has a subsequence that converges weakly a.e. (C_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly in L_{p}

has a subsequence that converges weakly a.e. (D_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly in mea-

sure has a subsequence that converges scalarly a.e. (E_{p}) Each L_{p}(𝔛)-bounded sequence of functions that converges scalarly in L_{p}

has a subsequence that converges scalarly a.e. For convenience, a schematic summary of the properties is given below, in which a double arrow indicates an implication that is always valid.

weakly a.e.  ⇒  ⟵^{A}  scalarly a.e.
c↑  e↗  ↖b  ↑d ⇓
scalarly in L_{p} ⇒  scalarly in measure .

Remarks. 1. Clearly, subsequential convergence is the most one can expect in passing from scalar in measure or scalar in L_{p} to scalar a.e. convergence.

2. Note that if 𝔛 has (Property_{p}) and p < q, then 𝔛 also has (Property_{q}).

3. Note the following obvious implications.

(A_{p}) ⟵ (B_{p}) ⟶ (C_{p})
↓  ↓
(D_{p}) ⟶ (E_{p})

4. For a fixed 1 ≤ p < ∞, a sequence {f_{n}} in L_{0}(𝔛) converges scalarly in L_{p} if and only if (i) {f_{n}} converges scalarly in measure and (ii) for each x^{∗} ∈ 𝔛^{∗}, the set {|x^{∗}f|^{p}} is uniformly integrable. Note that (ii) holds when {f_{n}} is L_{r}(𝔛)-bounded for some r > p.

3. L_{∞}(𝔛)-BOUNDED SEQUENCES

[3.i] From scalar convergence to weak a.e. convergence. In this subsection, we characterize those spaces having (A_{∞}), (B_{∞}), and (C_{∞}).

Theorem 3.1. Let 𝔛^{∗} have the Radon-Nikodým property. Then 𝔛 enjoys the following properties:

(1) Let {f_{n}} be a sequence in L_{0}(𝔛) which converges scalarly a.e. to some f ∈ L_{0}(𝔛). Then {f_{n}} converges weakly a.e. to f if and only if {f_{n}} is pointwisebounded a.e.

(2) Let {f_{n}} be a sequence in L_{0}(𝔛) which converges scalarly a.e. to some f ∈ L_{0}(𝔛). Then {f_{n}} has a subsequence which converges weakly a.e. to f if and only if {f_{n}} has a subsequence which is pointwise-bounded a.e.

(3) Let {f_{n}} be a sequence in L_{0}(𝔛) which converges scalarly in measure. Then {f_{n}} has a subsequence which converges weakly a.e. if and only if {f_{n}} has a subsequence which is pointwise-bounded a.e.

Proof. By Fact 1.4 each a.e. weakly convergent sequence is bounded a.e., and so necessity in (1)-(3) (for an arbitrary Banach space) is clear. To prove sufficiency observe that we may take f = 0 without loss of generality. We may also assume that 𝔛^{∗} is separable. (Indeed, by the Pettis measurability theorem [15], there is a separable subspace 𝔛_{0} of 𝔛 such that the f_{n}’s are essentially valued in 𝔛_{0}. Because 𝔛^{∗} has the RNP, 𝔛_{0}^{∗} must be separable [18, Theorem 2].) Let {x_{i}^{∗}} be dense in 𝔛^{∗}. Now we prove sufficiency in (1). Since {f_{n}} is scalarly null a.e., it follows that for each i there is a set A_{i} of full measure such that if ω ∈ A_{i}, then lim_{n} x_{i}^{∗}f_{n}(ω) = 0. Put A = ∩_{i}A_{i}. Since the f_{n}’s are pointwise-bounded on some set B of full measure, and since {x_{i}^{∗}} is dense in 𝔛^{∗}, it follows that lim_{n} x^{∗}f_{n}(ω) = 0 for each x^{∗} ∈ 𝔛^{∗} and for each ω ∈ A ∩ B. Thus, {f_{n}} is weakly null a.e. Sufficiency in (2) follows at once. Finally, we prove sufficiency in (3). By first passing to a subsequence we may assume that {f_{n}} is pointwise bounded almost everywhere. Since {f_{n}} converges to zero scalarly in measure, for each i the sequence {x_{i}^{∗}f_{n}}_{n} converges in measure to the null function. So by a Cantor diagonalization argument there exists a subsequence {f_{n_{k}}} such that for almost all ω

lim_{k} x_{i}^{∗} (f_{n_{k}} (ω)) = 0 .

for all i. Now, arguing as before, the pointwise boundedness implies that {f_{n_{k}}} is weakly null a.e.

For the Banach spaces ℓ_{1}, C(Δ), and the James tree space, Davis and Johnson [2] constructed examples of L_{∞}(𝔛)-bounded sequences that converge scalarly a.e. but not weakly a.e. They conjectured that such a sequence exists for any space 𝔛 whose dual fails the Radon-Nikodým property (RNP). Combined with work of Uhl [20] and Stegall [19], a result of Edgar [7] shows that their conjecture was correct. In fact, rather more can be said as the following theorem (whose proof was inspired by [7]) shows.

Theorem 3.2. For a Banach space 𝔛, the following are equivalent:

(1) 𝔛^{∗} has the Radon-Nikodým property (i.e. 𝔛 is an Asplund space);

(2) 𝔛 has (A′_{∞});

(3) 𝔛 has (A_{∞});

(4) 𝔛 has (B_{∞}).

Proof. Several implications follow from Theorem 3.1. To prove the other implications, suppose that 𝔛^{∗} fails the RNP. Then [20] there is a separable subspace 𝔛_{0} of 𝔛 such that 𝔛_{0}^{∗} is not separable. We shall construct an L_{∞}(𝔛)-bounded sequence {g_{n}} of 𝔛_{0}-valued functions such that g_{n} → 0 scalarly a.e. and scalarly in L_{r}(𝔛) for 1 ≤ r < ∞, but such that no subsequence of {g_{n}} converges weakly a.e. (This particular construction has been fruitful in several similar characterizations of 𝔛^{∗} having the RNP [5,7,8].) This will show that 𝔛 fails (A_{∞}), thus completing the proof of the theorem.

Let Δ = {−1, 1}^{ℕ} be the Cantor group with Haar measure ν. Let {Δ_{k}^{n}: k = 1, …, 2^{n}} be the standard n-th partition of Δ. Thus Δ_{1}^{0} = Δ and Δ_{k}^{n} = Δ_{2k−1}^{n+1} ∪ Δ_{2k}^{n+1} and ν(Δ_{k}^{n}) = 2^{−n}. Instead of our usual Lebesgue measure space on [0, 1], we shall now take our underlying measure space to be the completion of ν for the completion of the Borel σ-algebra of Δ. Thus L_{p}(𝔛) will denote L_{p}(Δ, ν; 𝔛).

We consider the space C(Δ) of real-valued continuous functions on Δ as a subspace of L_{∞}(Δ, ν). Let {1_{Δ}}∪{h_{k}^{n}: n = 0, 1, 2, … and k = 1, …, 2^{n}} be the usual Haar basis of C(Δ), where h_{k}^{n}: Δ → ℝ is given by

h_{k}^{n} = 1_{Δ_{2k−1}^{n+1}} − 1_{Δ_{2k}^{n+1}} .

Let {e_{k}^{n}: n = 0, 1, 2, … and k = 1, …, 2^{n}} be an enumeration (lexicographically) of the usual ℓ_{1} basis and let H: ℓ_{1} → L_{∞} be the Haar operator that takes e_{k}^{n} to h_{k}^{n}.

By Stegall’s Factorization Theorem [S, Theorem 4], H factors through 𝔛_{0}, i.e. there are bounded linear operators R: ℓ_{1} → 𝔛_{0} and S: 𝔛_{0} → L_{∞} such that H = SR.

ℓ_{1}  ⟶^{H}  L_{∞}(Δ) ⊃ C(Δ)
↘^{R}  ↗^{S}
𝔛_{0}

Let R̃ be the natural extension of R to a bounded linear operator from L_{1}(ℓ_{1}) to L_{1}(𝔛_{0}).

Consider the sequence {f_{m}} of L_{1}(ℓ_{1}) functions given by

f_{m}(·) = \frac{1}{m} ∑_{n=1}^{m} ∑_{k=1}^{2^{n}} h_{k}^{n}(·)e_{k}^{n} .

Let g_{m} = R̃(f_{m}). Clearly, {g_{m}} is L_{∞}(𝔛)-bounded since ||f_{m}(ω)||_{ℓ_{1}} = 1 for ν-a.e. ω and each m.

To examine the scalar behavior of {g_{m}}, note that if y^{∗} ∈ 𝔛_{0}^{∗}, then

y^{∗}g_{m}(·) = (R^{∗}y^{∗})f_{m}(·) ,

where R^{∗}y^{∗} ∈ ℓ_{1}^{∗}. So to show that {g_{m}} converges to the null function scalarly a.e. and scalarly in L_{r}(𝔛) we need only show the same for {f_{m}}. So fix a functional x^{∗} in ℓ_{1}^{∗}; let x^{∗} have the form (α_{k}^{n}) ∈ ℓ_{∞}, lexicographically ordered. Then

x^{∗}f_{m}(ω) = \frac{1}{m} ∑_{n=1}^{m} X_{n}(·)  where  X_{n}(·) = ∑_{k=1}^{2^{n}} h_{k}^{n}(·)α_{k}^{n} .

Note that ∥X_{n}∥_{∞} ≤ ∥x^{∗}∥, that each X_{n} has zero mean, and that ∫ X_{n}X_{m} dν = 0 when n ≠ m. The Strong Law of Large Numbers for uncorrelated random variables with uniformly bounded second moments [1, Theorem 5.1.2] gives that {x^{∗}f_{m}} converges to the null function a.e. Since ∥x^{∗}f_{m}∥_{∞} ≤ ∥x^{∗}∥ it also follows that {x^{∗}f_{m}} converges to the null function in L_{p} for 0 ≤ p < ∞.

We shall now show that no subsequence of {g_{m}} converges weakly a.e. Since {g_{m}} is scalarly null a.e., it suffices to show that no subsequence is weakly null a.e. For ω ∈ Ω, let ℓ_{ω} ∈ [C (Δ)]^{∗} be the point evaluation at ω functional and let ℓ̃_{ω} ∈ [L_{∞} (Δ)]^{∗} be any Hahn-Banach extension. Then

(S^{∗}ℓ̃_{ω}) (g_{n} (ω)) = ℓ_{ω} (Hf_{n} (ω)) = (\frac{1}{m} ∑_{n=1}^{m} ∑_{k=1}^{2^{n}} h_{k}^{n}(ω)h_{k}^{n}(·)) (ω) = 1 ,

and thus no subsequence of {g_{n}} converges weakly a.e. (to the null function) in L_{0}(𝔛_{0}).

Property C_{∞}, on the other hand, is a much weaker property according to the following mildly surprising result.

Proposition 3.3. Let {f_{n}} be a sequence in L_{0}(𝔛) that converges to the null function scalarly in L_{∞}. Then {f_{n}} is weakly null a.e. In particular, every Banach space enjoys (C_{∞}) (and a fortiori (E_{∞})).

Proof. For each n ≥ 1, we may write f_{n} = g_{n} + h_{n}, where h_{n} has L_{∞}(𝔛)-norm at most 1/n and g_{n} is countably-valued (see e.g. [3, II.1.3]). By choosing a suitable representative of g_{n} in L_{∞}(𝔛), we may express g_{n} as

g_{n} = ∑_{k=1}^{∞} x_{k}^{n} 1_{E_{k}^{n}},

where each E_{k}^{n} has strictly positive measure and, for each n, Ω is the disjoint union of {E_{k}^{n}}_{k}. Note that {g_{n}} also converges to the null function scalarly in L_{∞}(𝔛). Hence, for each ω ∈ Ω and each x^{∗} ∈ X^{∗}, we have

|x^{∗}(g_{n}(ω))| ≤ sup_{k} |x^{∗}(x_{k}^{n})| = ∥x^{∗}(g_{n})∥_{∞} → 0

as n → ∞. Hence {g_{n}(ω)} is weakly null for all ω ∈ Ω. Clearly, {h_{n}(ω)} is norm-null a.e., whence {f_{n}} is weakly null a.e.

Perhaps the following theorem is the most useful analogue for scalar convergence in general Banach spaces of the familiar fact from real analysis that each sequence that converges in measure has a subsequence that converges a.e.

Theorem 3.4. Let K be a weakly compact subset of a Banach space 𝔛 and let {f_{n}} be a sequence in L_{0}(𝔛) such that each f_{n} is essentially K-valued. If {f_{n}} converges scalarly in measure to f ∈ L_{0}(𝔛), then some subsequence {f_{n_{k}}} converges weakly a.e. to f. (In particular, f is essentially K-valued.)

Proof. Clearly, we may assume that 𝔛 is separable. First, we show that f(ω) ∈ K′ = \overline{conv(K)} a.e. Suppose this is not the case. Then there exists a closed ball B ⊂ 𝔛 \ K′ such that μ(A) > 0, where A = {ω : f(ω) ∈ B} > 0. By the Hahn-Banach Theorem there exists x^{∗} ∈ 𝔛^{∗} and α, β ∈ ℝ such that

sup_{k∈K′} x^{∗}(k) < α < β < inf_{b∈B} x^{∗}(b) .

Since each f_{n} is (without loss of generality) K-valued, it follows that

x^{∗}(f_{n}(ω)) < α < β < x^{∗}(f(ω))

for all n ≥ 1 and all ω ∈ A. This contradicts the fact that {f_{n}} converges scalarly in measure to f. Hence, by replacing f_{n} by f_{n} − f and K by K′ − K′, we may assume without loss of generality that {f_{n}} converges scalarly in measure to the null function and that K is a separable weakly compact set containing zero. It is easily seen that the weak topology on K is generated by a sequence {x_{n}^{∗}} in 𝔛^{∗}. By a Cantor diagonal argument there exists a subsequence {f_{n_{k}}} and a set Ω′ ⊂ Ω of full measure such that x_{n}^{∗}(f_{n_{k}}(ω)) → 0 as k → ∞ for all n ≥ 1 and for all ω ∈ Ω′ Now let x^{∗} ∈ 𝔛^{∗} and let ε > 0. There exists δ > 0 and N ≥ 1 such that

{k ∈ K : |x^{∗}(k)| < ε} ⊃ ∩_{i=1}^{N}{k ∈ K : |x_{i}^{∗}(k)| < δ} .

It follows that

{ω ∈ Ω : |x^{∗}(f_{n_{k}}(ω))| < ε for all sufficiently large k} ⊃ Ω′ ,

and so x^{∗}(f_{n_{k}}(ω)) → 0 as n → ∞ for all ω ∈ Ω′. Hence {f_{n_{k}}} is weakly null a.e.

Minor variations in the above proof gives the following result.

Theorem 3.5. Let K be a weakly compact subset of a Banach space 𝔛 and let {f_{n}} be a sequence in L_{0}(𝔛) such that each f_{n} is essentially K-valued. If {f_{n}} converges scalarly a.e. to f ∈ L_{0}(𝔛), then {f_{n}} converges weakly a.e. to f.

[3.ii] From scalar convergence in measure to scalar a.e. convergence. In the previous subsection, Proposition 3.3 shows that each Banach space enjoys (E_{∞}). In this subsection, we explore (D_{∞}).

For each 1 ≤ p ≤ ∞, it follows directly from the definitions that (B_{p}) implies (D_{p}); however, they are not equivalent. Indeed, Theorem 3.2 implies that ℓ_{1} fails (B_{p}) for each 1 ≤ p ≤ ∞. However, ℓ_{1} has (D_{p}) for each 1 ≤ p ≤ ∞, as the next theorem, which was pointed out to us by W.B. Johnson [12], shows. We are grateful to him for permission to include this result here.

Theorem 3.6. A scalarly null in measure sequence of L_{0}(ℓ_{1}) functions contains a scalarly null a.e. subsequence. In particular, ℓ_{1} satisfies properties (D_{p}) for each 1 ≤ p ≤ ∞.

The following lemma will be used in the proofs of several results, including Theorem 3.6.

Lemma 3.7. Let 𝔛 be a Banach space with a basis and {f_{n}} be a scalarly in measure null sequence of L_{0}(𝔛) functions. There exists a subsequence {f_{n_{k}}} of {f_{n}}, a blocking {𝔛_{k}} of the basis, and sequences {g_{k}} and {h_{k}} of L_{0}(𝔛) functions so that:

(i) {h_{k}} converges a.e (in 𝔛-norm) to the null function; and for each k:

(ii) f_{n_{k}} = g_{k} + h_{k}; (iii) g_{k} = P_{k}f_{n_{k}}, where P_{k} is the natural projection of 𝔛 onto 𝔛_{k}.

In particular, {g_{k}} is also scalarly null in measure.

Proof of Lemma 3.7. Let {x_{n}}_{n≥1} be a normalized basis for 𝔛 and let {x_{n}^{∗}} be the corresponding biorthogonal functionals. Consider a sequence {f_{n}}_{n≥1} of L_{0}(𝔛) functions that is scalarly null in measure. It suffices to construct inductively two increasing sequences {n_{k}}_{k≥1} and {m_{k}}_{k≥0} of integers and a sequence {g_{k}} of functions such that, for 𝔛_{k} ≡ sp {x_{i} : m_{k−1} < i ≤ m_{k}}, each g_{k} satisfies (iii) and

μ({ω : ∥g_{k}(ω) − f_{n_{k}}(ω)∥ ≥ 2^{−k}}) ≤ 2^{−k}.  (1)

To start the induction set m_{0} = 0 and n_{0} = 0. Suppose that k ≥ 1 and that n_{i} and m_{i} have been chosen for i ≤ k − 1. Since {f_{n}} is assumed to be scalarly null in measure, it follows that, for each fixed i, the sequence {x_{i}^{∗}(f_{n})}_{n} converges to zero in measure. So there exists n_{k} > n_{k−1} such that

μ ({ω : ∑_{i=1}^{m_{k−1}} |x_{i}^{∗}(f_{n_{k}})| ≥ 2^{−k−1}}) ≤ 2^{−k−1} .

Hence there exists m_{k} > m_{k−1} such that, for 𝔛_{k} ≡ sp {x_{i} : m_{k−1} < i ≤ m_{k}}, the function g_{k} as given in (iii) satisfies (1), which completes the induction.

Proof of Theorem 3.6. Consider a sequence {f_{n}} of L_{0}(ℓ_{1}) functions that is scalarly null in measure. Let {e_{n}} be the standard basis of ℓ_{1}. Find a blocking {𝔛_{k}} of {e_{n}} and sequences {g_{k}} and {h_{k}} as given by Lemma 3.7.

In view of (i), it is enough to show that {g_{k}} has a subsequence that is scalarly null a.e. With that in mind, we establish the following claim. Claim. Given ε > 0,

sup_{∥x^{∗}∥≤1}μ ({|x^{∗}(g_{k})| > ε}) → 0

as k → ∞. Proof of Claim. Suppose not. Then there exist ε > 0 and x_{k}^{∗} ∈ ℓ_{1}^{∗}, with ∥x_{k}^{∗}∥ ≤ 1, such that

lim sup_{k} μ({|x_{k}^{∗}(g_{k})| > ε}) > ε.

Note that ℓ_{1} = (∑ ⊕E_{k})_{1} and so ℓ_{1}^{∗} = (∑ ⊕E_{k}^{∗})_{∞}. Thus there exists x^{∗} ∈ ℓ_{1}^{∗} such that, for each k, the functionals x_{k}^{∗} and x^{∗} have identical restrictions to 𝔛_{k}^{∗}. Hence,

lim sup_{k} μ({|x^{∗}(g_{k})| > ε}) > ε,

which contradicts the fact that {g_{k}} is scalarly null in measure.

It follows from the claim that there exists a subsequence {g_{n_{k}}} such that

sup_{∥x^{∗}∥≤1}μ({|x^{∗}(g_{n_{k}})| > 2^{−k}}) < 2^{−k}.

Clearly {g_{n_{k}}} is scalarly null a.e.

However, we know of at least one space that fails (D_{∞}).

Theorem 3.8. C[0, 1] fails property (D_{∞}).

Proof. Consider a sequence {f_{n}} in the unit ball of L_{∞} (C [0, 1]). With Ω_{i} = [0, 1], write f_{n}(·) ≡ f_{n} (·, t) : Ω_{1} × Ω_{2} → ℝ so that

(i) f_{n}(s, ·): Ω_{2} → ℝ is in C [0, 1] for almost all s ∈ Ω_{1} (ii) |f_{n}(s, t)| ⩽ 1 for each t ∈ Ω_{2} for almost all s ∈ Ω_{1}.

Such a sequence {f_{n}} is scalarly null in measure if and only if

(iii) f_{n}(·, t): Ω_{1} → ℝ converges in measure to the null function for each t ∈ Ω_{2}. To see this, note that if t ∈ Ω_{2} is fixed, then (x_{t}^{∗}f_{n}) (·) = f_{n}(·, t) where x_{t}^{∗} ∈ (C [0, 1])^{∗} is the point evaluation at t functional. As for the reverse implication, assume that (iii) holds and let x^{∗} ≡ ν ∈ (C [0, 1])^{∗} be a finite regular positive Borel measure on B (Ω_{2}). It suffices to show that x^{∗}f_{n}(·) ≡ ∫_{Ω_{2}} f_{n}(·, t) dν(t) converges to the null function in μ-measure. Towards this, let λ = μ × ν be the corresponding product measure on the completion A of B (Ω_{1} × Ω_{2}). Then (iii) implies that f_{n}(·, ·): Ω_{1}×Ω_{2} → ℝ converges to the null function in λ-measure and hence (by (ii)) also in L_{1} (Ω_{1} × Ω_{2}, A, λ). Since

∫_{Ω_{1}} (∫_{Ω_{2}} |f_{n}(s, t)| dν(t)) dμ(s) = ∫∫_{Ω_{1}×Ω_{2}} |f_{n}(s, t)| dλ → 0

as n → ∞, it follows that the sequence {l_{n}} of L_{1} functions given by

l_{n}(·) ≡ ∫_{Ω_{2}} |f_{n}(·, t)| dν(t)

converges to the null function in μ-measure, which gives the result.

For each positive integer n, let ñ be its binary representation as a finite sequence of 0 and 1’s. For t ∈ Ω, let t_{3} be its unique (nonterminating) ternary expansion into 0, 1, and 2’s. For 1 ⩽ k ⩽ n, let Γ(k, n) be the collection of all k-tuples (n_{1}, n_{2}, …, n_{k}) of positive integers that satisfy 1 ⩽ n_{1} < n_{2} < … < n_{k−1} < n_{k} = n. For γ = (n_{1}, n_{2}, …, n_{k}) in Γ(k, n), let A_{γ} be the set of t ∈ Ω for which t_{3} is of the form

0. ñ_{1} 2 ñ_{2} 2 ñ_{3} 2 … ñ_{k−1} 2 ñ_{k} 2 …,

i.e.

A_{γ} = ( 0. ñ_{1} 2 ñ_{2} 2 … ñ_{k−1} 2 ñ_{k} 1 \overline{2} , 0. ñ_{1} 2 ñ_{2} 2 … ñ_{k−1} 2 ñ_{k} \overline{2} ] .

For technical reasons, consider the subset

Ã_{γ} = ( 0. ñ_{1} 2 ñ_{2} 2 … ñ_{k−1} 2 ñ_{k} 20 \overline{2} , 0. ñ_{1} 2 ñ_{2} 2 … ñ_{k−1} 2 ñ_{k} 220\overline{2} ]

of A_{γ} along with the corresponding unions

A_{k}^{n} = ⋃_{γ∈Γ(k,n)} A_{γ}  and  Ã_{k}^{n} = ⋃_{γ∈Γ(k,n)} Ã_{γ} .

The following properties of these sets will be used:

(1) A_{k_{1}}^{n} ∩ A_{k_{2}}^{n} = ∅ if k_{1} ≠ k_{2};

(2) if t ∈ ∩_{j}A_{k_{j}}^{n_{j}} for an increasing sequence {n_{j}}, then {k_{j}} is also (strictly) increasing;

(3) if {n_{k}}_{k} is an increasing sequence of positive integers and t_{3} = 0. ñ_{1} 2 ñ_{2} 2 ñ_{3} 2 …, then t ∈ Ã_{k}^{n_{k}} for each k.

For each admissible n and k, find a continuous function g_{k}^{n}: Ω_{2} → [0, 1] that is supported on A_{k}^{n} and takes the value 1 on Ã_{k}^{n}. Lexicographically order the dyadic interval {I_{k}}_{k⩾1} of [0, 1] and let h_{k}: Ω_{1} → [0, 1] be the indicator function of I_{k}. Define f_{n}: Ω_{1} × Ω_{2} → ℝ by

f_{n}(s, t) = ∑_{k=1}^{n} h_{k}(s)g_{k}^{n}(t) .

Clearly, the corresponding sequence {f_{n}} is in the unit ball of L_{∞} (C [0, 1]) and it satisfies conditions (i), (ii), and, by (1) and (2), also (iii). Thus {f_{n}} converges scalarly in measure to the null function.

However, for any subsequence {n_{k}}_{k} of the positive integers, for the corresponding point t_{3} = 0. ñ_{1} 2 ñ_{2} 2 ñ_{3} 2 …, it follows from condition (3) that f_{n_{k}}(s, t) = h_{k}(s), which does not go pointwise a.e. to the null function.

We shall prove below (Corollary 4.11) that L_{1} fails (D_{p}) for 1 ≤ p < ∞, but we do not know what happens when p = ∞.

Question 3.9. Does L_{1} enjoy (D_{∞})?

This question can be reformulated as a question about functions of two variables as follows. For n ≥ 1, let f_{n}(s, t) be real-valued functions on the unit square which satisfy the following:

(i) f_{n}(s, t) = ∑_{k=1}^{N(n)} 1_{E_{n,k}}(s)g_{n,k}(t), where {E_{n,k}}_{k} is a partition of [0, 1] into sets of strictly positive measure;

(ii) ∫ |g_{n,k}(t)| dμ(t) ≤ 1 for all 1 ≤ k ≤ N(n); (iii) F_{n}(s) = ∫_{A} f_{n}(s, t)dμ(t) → 0 in measure as n → ∞ for every measurable

set A ⊂ [0, 1].

Question 3.9 paraphrased. For {f_{n}(s, t)} as above, does there always exist a subsequence {f_{n_{k}}(s, t)} such that ∫_{A} f_{n_{k}}(s, t)dμ(t) → 0 a.e. for each A ∈ Σ?

4. L_{p}(𝔛)-BOUNDED SEQUENCES

We now investigate what happens when L_{∞}(𝔛)-boundedness is weakened to L_{p}(𝔛)-boundedness.

[4.i] From scalar convergence to weak a.e. convergence. In this subsection, we shall use Dvoretzky’s theorem on the existence of almost spherical sections [6] to prove that for 1 ≤ p < ∞ none of the properties (A_{p}), (B_{p}) nor (C_{p}) can hold in any infinite-dimensional Banach space.

Let us first recall the q-Pettis norm of an L_{0}(𝔛) function (which might be infinite):

∥f∥_{P_{q}(𝔛)} = sup_{x^{∗}∈B(𝔛^{∗})} (∫_{Ω} |x^{∗}f(ω)|^{q} dμ)^{1/q} ,

for 1 ≤ q < ∞. The building block used in our construction is the basic example of [5] which we now recall (and refine slightly) for the reader’s convenience.

Proposition 4.1. Let 𝔛 be an infinite-dimensional Banach space and let E be a measurable subset of Ω. Given ε > 0, there exists f ∈ L_{∞}(𝔛) such that ∥f(·)∥ = 1_{E}(·) and ∥f∥_{P_{q}(𝔛)} < 2ε^{1/q} for each 1 ≤ q < ∞.

Proof. Since q → ∥f∥_{P_{q}(𝔛)} is an increasing function for a fixed f ∈ L_{0}(𝔛), it suffices to consider only 2 ≤ q < ∞. First we prove the result for E = Ω.

Let

{I_{k}^{n} = [(k − 1)/2^{n}, k/2^{n}) : n ≥ 0 and 1 ≤ k ≤ 2^{n}}

be the collection of dyadic subintervals of Ω. By Dvoretzky’s Theorem there exist unit vectors {e_{k}^{n}}_{k=1}^{2^{n}} in 𝔛 such that

\frac{1}{2} (∑_{k=1}^{2^{n}} |a_{k}^{n}|^{2})^{1/2} ≤ ∥∑_{k=1}^{2^{n}} a_{k}^{n}e_{k}^{n}∥ ≤ 2 (∑_{k=1}^{2^{n}} |a_{k}^{n}|^{2})^{1/2}  (1)

for all real numbers a_{k}^{n}. Define f_{n} : Ω → 𝔛 by

f_{n}(ω) = ∑_{k=1}^{2^{n}} 1_{I_{k}^{n}}(ω)e_{k}^{n}.

Note that ∥f_{n}(ω)∥ = 1 for all ω ∈ Ω. Fix x^{∗} ∈ B(𝔛^{∗}). Then (1) implies that (∑_{k=1}^{2^{n}} |x^{∗}(e_{k}^{n})|^{2})^{1/2} ≤ 2. Thus, for q ≥ 2, we have

∫_{Ω} |x^{∗}f_{n}(ω)|^{q} dμ = ∫_{Ω} |∑_{k=1}^{2^{n}} x^{∗}(e_{k}^{n})1_{I_{k}^{n}}(ω)|^{q} dμ

= 2^{−n} ∑_{k=1}^{2^{n}} |x^{∗}(e_{k}^{n})|^{q}

≤ 2^{−n} (∑_{k=1}^{2^{n}} |x^{∗}(e_{k}^{n})|^{2})^{q/2} ≤ 2^{q−n} ,

and so ∥f_{n}∥_{P_{q}(𝔛)} ≤ 2 · (2^{−n})^{\frac{1}{q}}, which gives the result.

An analogous construction can be carried out in any set E of positive measure, and the result is trivial anyhow for a set E of measure zero.

Now we beef up the previous result.

Proposition 4.2. Let 𝔛 be an infinite-dimensional Banach space and let h be a non-negative countably-valued measurable function defined on Ω. Given ε > 0 and 1 ≤ q_{0} < ∞ there exists f ∈ L_{0}(𝔛) with the following properties:

(1) ∥f(·)∥ = h(·);

(2) ∥f∥_{P_{q}(𝔛)} < ∞ for each 1 ≤ q < ∞;

(3) ∥f∥_{P_{q_{0}} (𝔛)} < ε.

Proof. Write h = ∑_{k=1}^{∞} a_{k}1_{E_{k}}, where the a_{k}’s are positive numbers and the E_{k}’s are disjoint measurable sets. Select positive numbers {ε_{k}} such that ∑_{k=1}^{∞} a_{k}^{q}ε_{k} is finite for each 1 ≤ q < ∞ and ∑_{k=1}^{∞} a_{k}^{q_{0}}ε_{k} < (ε/2)^{q_{0}}. By Proposition 4.1, for each k there exists f_{k} ∈ L_{∞}(𝔛) such that ∥f_{k}(·)∥ = 1_{E_{k}}(·) and ∥f_{k}∥_{P_{q}(𝔛)} < 2ε_{k}^{1/q}. Clearly, f = ∑_{k=1}^{∞} a_{k}f_{k} has the required properties.

Theorem 4.3. Let 𝔛 be an infinite-dimensional Banach space and let g be any non-negative measurable function that is not essentially-bounded. There exists a sequence {g_{n}} in L_{0}(𝔛) such that the following hold:

(1)

μ({ω : ∥g_{n}(ω)∥ > t}) ≤ μ({ω : g(ω) > t})

for all n ≥ 1 and for all t > 0; (2)

∑_{n=1}^{∞} ∥g_{n}∥_{P_{q}(𝔛)} < ∞

for each 1 ≤ q < ∞;

(3) {g_{n}} converges scalarly a.e. to the null function;

(4) for each subsequence {g_{n_{j}}} there exists a set A ⊂ Ω of full measure such

that

lim sup_{j} ∥g_{n_{j}}(ω)∥ = ∞

for each ω ∈ A.

In particular, no subsequence of {g_{n}} converges weakly on any set of strictly positive measure.

Proof. Let h be a non-negative countably-valued measurable function on Ω which is not essentially bounded and which satisfies h(ω) ≤ g(ω) for ω ∈ Ω. Use Proposition 4.2 to construct a sequence {g_{n}} of independent 𝔛-valued random variables such that

(i) each ∥g_{n}∥ has the same distribution as h, (ii) ∥g_{n}∥_{P_{q}(𝔛)} is finite for each 1 ≤ q < ∞, and (iii) ∥g_{n}∥_{P_{n}(𝔛)} ≤ 2^{−n} .

Clearly (1) is satisfied. Condition (2) follows from the observation that, if N ∈ ℕ and 1 ≤ q ≤ N, then by (ii) and (iii)

∑_{n=1}^{∞} ∥g_{n}∥_{P_{q}(𝔛)} ≤ ∑_{n=1}^{N} ∥g_{n}∥_{P_{q}(𝔛)} + ∑_{n=N+1}^{∞} ∥g_{n}∥_{P_{n}(𝔛)}

≤ ∑_{n=1}^{N} ∥g_{n}∥_{P_{q}(𝔛)} + ∑_{n=N+1}^{∞} 2^{−n} < ∞ .

Clearly, (3) follows from (2) using Fact 1.5. To prove (4), fix a subsequence {g_{n_{j}}}. Then, for each M > 0,

∑_{j=1}^{∞} μ({ω : ∥g_{n_{j}}(ω)∥ > M}) = ∑_{j=1}^{∞} μ({ω : h(ω) > M}) = ∞

since h does not belong to L_{∞}. So by the Borel-Cantelli lemma ∥g_{n_{j}}(ω)∥ > M infinitely often a.e.

An appropriate choice of the measurable function g (e.g. g(ω) = | log ω|) in Theorem 4.3 yields the following corollary.

Corollary 4.4. Let 𝔛 be a Banach space and let 1 ≤ p < ∞. The following are equivalent:

(1) 𝔛 is finite-dimensional;

(2) 𝔛 satisfies (A′_{p});

(3) 𝔛 satisfies (A_{p});

(4) 𝔛 satisfies (B_{p});

(5) 𝔛 satisfies (C_{p}).

Remarks. 1. Theorem 4.3 shows that there is no analogue for scalar convergence of the uniform boundedness principle: if 𝔛 is infinite-dimensional then ‘scalar boundedness a.e.’ does not imply ‘norm-boundedness a.e.’

2. If ∥f_{n}∥_{L_{q}(𝔛)} → 0 then clearly some subsequence is 𝔛-norm-null a.e. However, condition (2) of Theorem 4.3 suggests that searching for a non-trivial scalar integrability condition which implies weak a.e. convergence is probably futile.

[4.ii] From scalar convergence in measure or in L_{p} to scalar a.e. convergence. In this subsection we examine the properties (D_{p}) and (E_{p}) more closely for 1 ≤ p < ∞.

First we recall some notation from [16]. Let {x_{n}} be a basic sequence in a Banach space 𝔛 with coefficient functional sequence {x_{n}^{∗}} in 𝔛^{∗}. A family {𝔛_{n}} of finite-dimensional subspaces of [x_{n}] is a blocking of {x_{n}} provided there exists an increasing sequence of integers {n_{k}} with n_{1} = 1 such that 𝔛_{k} = [x_{i}]_{i=n_{k}}^{n_{k+1}−1} for each k. For 1 ≤ p ≤ ∞, if there is a positive constant c so that, for each collection of vectors {y_{i}}_{i=1}^{n}, where y_{i} ∈ 𝔛_{i},

∥∑_{i=1}^{n} y_{i}∥ ≤ c ∥ (∥y_{i}∥) ∥_{ℓ_{p}} ,

resp.

c ∥ (∥y_{i}∥) ∥_{ℓ_{q}} ≤ ∥∑_{i=1}^{n} y_{i}∥ ,

then we say that the blocking {𝔛_{k}} satisfies an upper (resp. lower) p-estimate.

Theorem 4.5. Fix 1 < q ≤ ∞ and let q′ be its conjugate exponent. Suppose that 𝔛 has a basis {x_{n}} with the property that each blocking of this basis satisfies an upper q-estimate. Then 𝔛 enjoys (D_{p}) for each q′ ≤ p ≤ ∞.

Proof. Wlog, p = q′. Fix a sequence {f_{n}} in B(L_{p}(𝔛)) that is scalarly null in measure. We need to extract an scalarly null a.e. subsequence. To this end, let {𝔛_{k}}, {P_{k}}, {g_{k}}, and {h_{k}} be as provided from Lemma 3.7. It suffices to show that {g_{k}} converges to the null function scalarly a.e.

Fix x^{∗} ∈ 𝔛^{∗} and let x_{k}^{∗} = x^{∗} ∘ P_{k} ∈ X^{∗}. Note that

|x^{∗}g_{k}(ω)| ≤ ∥x_{k}^{∗}∥ ∥g_{k}(ω)∥_{𝔛} .

Thus, for ε > 0 fixed

μ(\{|x^{∗}g_{k}| ≥ ε\}) ≤ [\frac{∥x_{k}^{∗}∥}{ε} ∥g_{k}∥_{L_{p}}]^{p} .

Note that each ∥g_{k}∥_{L_{p}} is bounded above by 2K where K is the basis constant of {x_{n}}. Thus

∑_{k=1}^{∞} μ(\{|x^{∗}g_{k}| ≥ ε\}) ≤ [\frac{2K}{ε} ]^{p} ∑_{k=1}^{∞} ∥x_{k}^{∗}∥^{p} .

Since the blocking {𝔛_{k}} satisfies an upper q-estimate (say with constant C),

∥ (∥x_{k}^{∗}∥) ∥_{ℓ_{p}} ≤ 2CK lim inf_{n} ∥∑_{k=1}^{n} x_{k}^{∗}∥ ≤ 2CK^{2} ∥x^{∗}∥ .

Thus ∑_{k} μ({|x^{∗}g_{k}| ≥ ε}) < ∞. So by Borel-Cantelli, {x^{∗}g_{k}} converges to the null function, as needed.

Minor variations in the above proof give that Theorem 4.5 remains true if the word basis is replaced by finite-dimensional decomposition.

However, there are many spaces that fail (E_{p}) (and hence fail (D_{p})).

Theorem 4.6. Fix 1 < q ≤ ∞ and let q′ be its conjugate exponent. Suppose that 𝔛 contains a weakly null semi-normalized basic sequence {x_{n}} which satisfies a lower q-estimate. Then 𝔛 fails (E_{p}) for each 1 ≤ p < q′.

Proof. We may assume without loss of generality that 𝔛 = [x_{n}]. Fix p ∈ [1, q′) and choose q_{0} ∈ (\frac{1}{q′} , \frac{1}{p}). Let \{g_{n}\} be a sequence of i.i.d. random variables defined on Ω with the same distribution as g_{0}(t) = t^{−q_{0}}. Define f_{n}: Ω → 𝔛 by

f_{n}(·) = g_{n}(·) x_{n} .

Since g_{0} ∈ L_{p} and since {x_{n}} is semi-normalized, {f_{n}} is an L_{p}(𝔛)-bounded sequence. For each x^{∗} ∈ 𝔛^{∗},

∥x^{∗}f_{n}∥_{L_{p}} = |x^{∗}(x_{n})| ∥g_{n}∥_{L_{p}} .

Thus {f_{n}} converges scalarly in L_{p} to the null function.

Fix a subsequence {f_{n_{j}}} of {f_{n}}. It suffices to show that {f_{n_{j}}} is not scalarly null a.e. To this end, let {x_{n}^{∗}} be the sequence of biorthogonal functionals satisfying x_{n}^{∗}(x_{m}) = δ_{nm}. Since {x_{n}} is semi-normalized and satisfies a lower q-estimate, it follows that {x_{n}^{∗}} satisfies an upper q′-estimate. Consider the element x^{∗} ∈ 𝔛^{∗} given by x^{∗} = ∑_{j} j^{−q_{0}} x_{n_{j}}^{∗}, which converges in 𝔛^{∗} since {x_{n}^{∗}} satisfies an upper q′-estimate. Fix M > 0. Since

μ\{x^{∗}f_{n_{j}} > M\} = μ\{j^{−q_{0}}g_{n_{j}} > M\} = [Mj^{q_{0}}]^{− \frac{1}{q_{0}}} = M^{− \frac{1}{q_{0}}} j^{−1} ,

we see that ∑_{j} μ{x^{∗}f_{n_{j}} > M} = ∞, and so by the Borel-Cantelli lemma there is a set A of full measure such that if ω ∈ A then |x^{∗}f_{n_{j}}(ω)| > M infinitely often. Thus this subsequence does not converge scalarly a.e.

By a theorem of Prus [Pr] every nearly uniformly convex space (see [H] for the definition of this property) satisfies the hypothesis of Theorem 4.6 for some 1 < q < ∞ and so we obtain the following corollary.

Corollary 4.7. Suppose that 𝔛 contains a super-reflexive or (more generally) a nearly uniformly convexifiable infinite-dimensional subspace. Then 𝔛 fails (E_{p}) for some 1 < p < ∞.

The following two corollaries follow from the previous two theorems by considering the standard unit vector basis of ℓ_{p} and c_{0}.

Corollary 4.8. Fix 1 < q < ∞ and let q′ be its conjugate exponent. Then ℓ_{q} has (D_{p}) (resp. (E_{p})) if and only if 1 < q′ ≤ p ≤ ∞.

Corollary 4.9. c_{0} has (D_{p}) (and thus (E_{p})) for each 1 ≤ p ≤ ∞.

Note that in Corollary 4.8, if q ↓ 1 then p ↑ ∞, which suggests that ℓ_{1} should fail D_{p} for each 1 ≤ p < ∞. Strangely, however, the truth is the complete opposite as was proved in Theorem 3.6 above: ℓ_{1} has D_{p} for all 1 ≤ p ≤ ∞.

Finally, we determine the range of values of p for which L_{q} satisfies (D_{p}).

Corollary 4.10. Fix 1 < q < ∞ and let q′ be its conjugate exponent. Then L_{q} has (D_{p}) (resp. (E_{p})) if and only if max (2, q′) ≤ p ≤ ∞.

Proof. Since ℓ_{2} and ℓ_{q} each embed into L_{q}, it follows from Corollary 4.8 that L_{q} fails (E_{p}) for 1 ≤ p < max (2, q′). Since L_{q} has type min(2, q) and the Haar system H forms an unconditional basis for L_{q}, each blocking of H satisfies an upper min(2, q)- estimate. So by Theorem 4.5, if p = max (2, q′) then L_{q} enjoys (D_{p}).

Let H_{1} denote the Hardy space of analytic functions on the unit disk in the complex plane with the usual L_{1} norm (see e.g. [18]). It is known that H_{1} contains subspaces that are isomorphic to ℓ_{q} for 1 ≤ q ≤ 2 (see e.g. [4]). Hence Corollary 4.8 implies the next result (cf. Question 3.9).

Corollary 4.11. H_{1} (and therefore also L_{1}) fails (E_{p}) for 1 ≤ p < ∞.

Remark. We do not know of a reflexive space that satisfies (D_{1}).

5. Completeness

In this section we prove some completeness results for the topologies of scalar convergence considered in this paper. First we recall the appropriate definitions. Let E be a topological vector space. A sequence {x_{n}} in E is a Cauchy sequence if for every zero-neighborhood U there exists N ≥ 1 such that x_{n} − x_{m} ∈ U for all n, m ≥ N. We shall say that E is complete if every Cauchy sequence converges.

Theorem 5.1. Let 𝔛 be an infinite-dimensional Banach space. The topologies of scalar convergence in measure and scalar convergence in L_{p} ( 1 ≤ p < ∞) are incomplete.

Proof. This result is very similar in spirit to the fact that the Pettis norm is incomplete whenever 𝔛 is infinite-dimensional [11]. We refer the reader to the proof of the incompleteness of the Pettis norm that is given in [5]. The construction there, which utilizes Dvoretzky’s Theorem on almost spherical sections [6], can easily be modified, using the estimates of Proposition 4.1, to construct a sequence of functions that is Cauchy but not convergent in the topology of scalar convergence in measure and L_{p} for 1 ≤ p < ∞.

Theorem 5.2. The topology of scalar convergence in L_{∞} is complete if and only if 𝔛 is weakly sequentially complete.

Proof. First suppose that 𝔛 is not weakly sequentially complete. Let {x_{n}} be a weak Cauchy sequence that does not converge weakly and let f_{n}(ω) = x_{n} (n ≥ 1). Clearly, {f_{n}} is a Cauchy sequence in the topology of scalar convergence in L_{∞}. By Proposition 3.3, a limit of this sequence, say f, would have to satisfy f(ω) = weak-lim f_{n}(ω) almost everywhere. Hence {f_{n}} does not converge.

For the converse, suppose that 𝔛 is weakly sequentially complete. Let {f_{n}} be a Cauchy sequence in the topology of scalar convergence in L_{∞}. By adapting the proof of Proposition 3.3, we see that the weak sequential completeness of 𝔛 guarantees that there exists a function f : Ω → 𝔛 such that f(ω) = weak-lim f_{n}(ω) a.e. By the Pettis measurability theorem [P] and Proposition 1.1, f is stronglymeasurable, i.e., f ∈ L_{0}(𝔛). It now follows easily from the fact that {f_{n}} is a Cauchy sequence that {f_{n}} converges to f scalarly in L_{∞}.

Of more relevance to this paper is the convergence of a pointwise-bounded or an L_{p}(𝔛)-bounded Cauchy sequence. We investigate this question next for the topology of scalar convergence in measure. For brevity’s sake we shall say that a sequence is “scalarly Cauchy in measure” if it is a Cauchy sequence for the topology of scalar convergence in measure.

Theorem 5.3. Let 𝔛 be a reflexive Banach space. Then each pointwise-bounded sequence {f_{n}} in L_{0}(𝔛) that is scalarly Cauchy in measure converges scalarly in measure.

Proof. We may assume that 𝔛 is separable and hence that 𝔛^{∗} is separable. Arguing now as in Proposition 3.1 there exists a subsequence {f_{n_{k}}} such that {f_{n_{k}}(ω)} is a weakly Cauchy sequence in 𝔛 almost everywhere. Since 𝔛 is reflexive it is weakly sequentially complete and so (by the Pettis Measurability Theorem and Proposition 1.1) there exists f in L_{0}(𝔛) such that f_{n_{k}} converges to f weakly a.e., thus also scalarly in measure, which is enough.

Theorem 5.4. Let 𝔛 be a reflexive Banach space. Then each L_{1}(𝔛)-bounded sequence {f_{n}} that is scalarly Cauchy in measure converges scalarly in measure.

Proof. First, we may assume that 𝔛 is separable. By a deep result of Zippin [21] every separable reflexive Banach space is isomorphic to a closed subspace of a reflexive Banach space with a basis. So we may assume that 𝔛 is isomorphically embedded into a reflexive Banach space 𝔜 with a basis. Clearly, {f_{n}} is scalarly Cauchy in measure when viewed as a sequence in L_{0}(𝔜). By Proposition 1.1, it suffices to show that {f_{n}} converges to some f in L_{0}(𝔜).

Since 𝔜 is reflexive, a normalized basis {e_{k}} for 𝔜 is both boundedly complete and shrinking [14]. Let C be the basis constant of {e_{k}}. For each n, we can expand f_{n} with respect to the basis {e_{k}} thus:

f_{n}(ω) = ∑_{k} f_{n,k}(ω)e_{k}.

For each k, the sequence {f_{n,k}}_{n} is Cauchy in measure, and hence converges in measure to some g_{k} ∈ L_{0}. By Fatou’s Lemma, we have

∫_{Ω} sup_{N} ∥∑_{k=1}^{N} g_{k}(ω)e_{k}∥ dμ ≤ C ∫_{Ω} lim inf_{N→∞} ∥∑_{k=1}^{N} g_{k}(ω)e_{k}∥ dμ

≤ C lim inf_{N→∞} ∫_{Ω} ∥∑_{k=1}^{N} g_{k}(ω)e_{k}∥ dμ

≤ C lim inf_{N→∞} (lim inf_{n→∞} ∫_{Ω} ∥∑_{k=1}^{N} f_{n,k}(ω)e_{k}∥ dμ)

≤ C lim inf_{N→∞} (C lim inf_{n→∞} ∫_{Ω} ∥∑_{k=1}^{∞} f_{n,k}(ω)e_{k}∥ dμ)

≤ C^{2} sup_{n} ∥f_{n}∥_{L_{1}(𝔜)} < ∞.  (1)

Hence

sup_{N} ∥∑_{k=1}^{N} g_{k}(ω)e_{k}∥ < ∞ a.e.

Since {e_{n}} is boundedly complete it follows that f(·) ≡ ∑_{k=1}^{∞} g_{k}(·)e_{k} is in L_{0}(𝔛); moreover, it follows from (1) that f ∈ L_{1}(𝔜).

Fix y^{∗} ∈ 𝔜^{∗} and N ≥ 1. Clearly,

y^{∗} (∑_{k=1}^{N} f_{n,k}e_{k}) → y^{∗} (∑_{k=1}^{N} g_{k}e_{k})  (2)

in measure as n → ∞. Let α_{n} denote the norm of the restriction of y^{∗} to [e_{k}]_{k≥n}. Since {e_{n}} is a shrinking basis, α_{n} → 0 as n → ∞. Now

∫ |y^{∗} ( ∑_{k=N+1}^{∞} f_{n,k}e_{k})| dμ ≤ α_{N+1} ∫ ∥ ∑_{k=N+1}^{∞} f_{n,k}e_{k}∥ dμ
≤ α_{N+1}(1 + C) sup_{n} ∥f_{n}∥_{L_{1}(𝔜)} → 0  (3)

as N → ∞. Combining (2) and (3) we see that {f_{n}} converges to f scalarly in measure.

The proof of Theorem 5.3 apparently uses only the weak sequential completeness of 𝔛 and the fact that 𝔛^{∗} has the RNP. However, by Rosenthal’s ℓ_{1} theorem [17], these two properties are equivalent to 𝔛 being reflexive. Clearly, a necessary condition for the conclusion of Theorems 5.3 and 5.4 to hold is that 𝔛 is weakly sequentially complete, and when 𝔛 has an unconditional basis this condition is also sufficient, as our next two results show. However, we have not been able to determine general necessary and sufficient conditions on 𝔛 so that the conclusions of Theorems 5.3 and 5.4 hold. In view of the next two theorems, which establish the desired conclusions for ℓ_{1}, it is clear that the method of proof of Theorem 3.2 will not be of use in this situation.

Theorem 5.5. Let 𝔛 be a weakly sequentially complete Banach space with an unconditional basis. Then each pointwise-bounded sequence {f_{n}} in L_{0}(𝔛) that is scalarly Cauchy in measure converges scalarly in measure.

Proof. Let {e_{n}} be a normalized unconditional basis for 𝔛. We may assume, without loss of generality, that

∥∑_{n} a_{n}e_{n}∥ ≤ ∥∑_{n} b_{n}e_{n}∥  (1)

whenever |a_{n}| ≤ |b_{n}| for all n. The fact that 𝔛 is weakly sequentially complete implies that {e_{n}} is boundedly complete [14].

By assumption,

sup_{n} ∥f_{n}(ω)∥ = M(ω) < ∞ a.e. .  (2)

Also, for each n, we can expand f_{n} with respect to the basis {e_{k}} thus:

f_{n}(ω) = ∑_{k} f_{n,k}(ω)e_{k} .

For each k, the sequence {f_{n,k}}_{n} is a Cauchy sequence in the topology of convergence in measure, and hence converges in measure to some g_{k}. Now (1) and (2) imply that

sup_{N} ∥∑_{k=1}^{N} g_{k}(ω)∥ ≤ M(ω) a.e. .

Since {e_{n}} is boundedly complete it follows that f(·) = ∑_{n=1}^{∞} g_{n}(·)e_{n} is in L_{0}(𝔛). Let h_{n} = f − f_{n}, and so

h_{n}(·) = ∑_{k}(g_{k} − f_{n,k})(·) e_{k} .

To complete the proof of the theorem, it suffices to show that {h_{n}} is scalarly null in measure.

So suppose, to derive a contradiction, that {h_{n}} is not scalarly null in measure. Then there exists x^{∗} ∈ S(𝔛^{∗}) and ε > 0 such that

μ{ω : |x^{∗}h_{n}(ω)| > ε} > ε  (3)

for infinitely many n.

The gliding hump argument of Lemma 3.7 yields a subsequence {h_{n_{k}}}_{k} and a blocking {𝔛_{k}} of the basis such that each h_{n_{k}} satisfies (3) and

μ{ω : ∥h_{n_{k}}(ω) − P_{k}h_{n_{k}}(ω)∥ > ε/4} < ε/4,  (4)

where P_{k} is the natural projection of 𝔛 onto 𝔛_{k}. We may define y^{∗} ∈ 𝔛^{∗} by defining its action on each x_{k} ∈ 𝔛_{k}:

y^{∗}(x_{k}) = 0 (k odd); y^{∗}(x_{k}) = x^{∗}(x_{k}) (k even) .  (5)

Then by (1), we have ∥y^{∗}∥ ≤ ∥x^{∗}∥ ≤ 1, and so from (3), (4) and (5) we deduce that

μ{ω : |y^{∗}h_{n_{k}}(ω)| > ε/4} < ε/4 (k odd)  (6)

while

μ{ω : |y^{∗}h_{n_{k}}(ω)| > ε/2} > 3ε/4 (k even).  (7)

Clearly, (6) and (7) contradict the fact that {h_{n}} is scalarly Cauchy in measure

Theorem 5.6. Let 𝔛 be a weakly sequentially complete Banach space with an unconditional basis. Then each L_{1}(𝔛)-bounded sequence {f_{n}} that is scalarly Cauchy in measure converges scalarly in measure.

Proof. Let {e_{k}} be a normalized unconditional basis for 𝔛. Let

f_{n}(ω) = ∑_{k} f_{n,k}(ω)e_{k}

be the expansion of f_{n}. Now, arguing as in the first half of Theorem 5.4, it can be shown that, for each k, f_{n,k} → g_{k} in measure as n → ∞ and that f(·) ≡ ∑_{k} g_{k}(·)e_{k} belongs to L_{1}(𝔛). Now, arguing as in second half of Theorem 5.5, one uses the unconditionality of {e_{k}} to prove that {f_{n}} converges to f scalarly in measure.

Remark. Note that Theorems 5.5 and 5.6 apply to both ℓ_{1} and H_{1}.

Finally, straightforward modifications to the proofs of Theorems 5.4 and 5.6 yield the following.

Theorem 5.7. Fix 1 ≤ p < ∞. Let 𝔛 be a weakly sequentially complete Banach space that is either reflexive or has an unconditional basis. Then each L_{p}(𝔛)- bounded sequence that is scalarly Cauchy in L_{p} converges scalarly in L_{p}.

Scalar Convergence 25

References

1. Kai Lai Chung, A Course in Probability Theory, Probability and Mathematical Statistics, vol. 21, Academic Press, New York, 1974.

2. W. J. Davis and W. B. Johnson, Weakly Convergent Sequences of Banach Space Valued Random Variables, Banach Spaces of Analytic Functions (J. Baker, C. Cleaver, and J. Diestel, eds.), Lecture Notes in Math., vol. 604, Springer-Verlag, New York-Berlin, 1977, pp. 29–31.

3. J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R.I., 1977.

4. S.J. Dilworth, Intersection of Lebesgue spaces L_{1} and L_{2}, Proc. Amer. Math. Soc. 103 (1988), 1185–1188.

5. S.J. Dilworth and Maria Girardi, Bochner vs. Pettis norms: examples and results, Banach Spaces (Bor-Luh Lin and William B. Johnson, ed.), Contemp. Math., vol. 144, American Mathematical Society, Providence, Rhode Island, 1993, pp. 69–80.

6. A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Symp. on Linear Spaces, Jerusalem, 1961, pp. 123–160.

7. G. A. Edgar, Asplund Operators and A. E. Convergence, J. Multivariate Anal. 10 (1980), 460–466.

8. N. Ghoussoub and P. Saab, Weak Compactness in Spaces of Bochner Integrable Functions and the Radon-Nikodým property, Pacific J. Math. 110 (1984), no. 1, 65–70.

9. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743–749.

10. R. C. James, Super-reflexive spaces with bases, Pacific J. Math 41 (1972), 409–419.

11. Liliana Janicka and Nigel J. Kalton, Vector Measures of Infinite Variation, Bull. Polish Acad. Sci. Math. XXV (1977), no. 3, 239–241.

12. W.B. Johnson, Personal Communication.

DILWORTH and GIRARDI 26

13. Michel Ledoux and Michel Talagrand, Probability in Banach Spaces, Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1991.

14. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer-Verlag, Berlin-Heidelberg, 1977.

15. B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277– 304.

16. Stanislaw Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. 7 (1989), 507–521.

17. H.P. Rosenthal, A characterization of Banach spaces containing ℓ_{1}, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413.

18. Walter Rudin, Real and complex Analysis, McGraw-Hill, New York, 1974.

19. Charles Stegall, The Radon-Nikodým Property in Conjugate Banach Spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223.

20. J. J. Uhl, Jr., A note on the Radon-Nikodým property for Banach spaces, Rev. Roumaine Math. Pures Appl. 17 (1972), 113–115.

21. M. Zippin, Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371–379.