FROM WEAK TO STRONG TYPES OF L_{E}^{1}-CONVERGENCE BY THE BOCCE-CRITERION

Erik J. Balder Maria Girardi^{∗} Vincent Jalby^{∗∗}

14 February 1994

Studia Math. 111 (1994) 241–262

Abstract. Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space L_{E}^{1} to be norm convergent (resp. relatively norm compact), thus extending the known results for L_{ℝ}^{1}. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in L_{E}^{1}. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in L_{E}^{1} are also studied.

1. Introduction

Vaguely speaking, a relatively weakly compact set in L_{ℝ}^{1} is relatively norm compact if the functions in the set do not oscillate too much. Specifically, a relatively weakly compact subset of L_{ℝ}^{1} is relatively norm compact if and only if it satisfies the Bocce criterion (an oscillation condition) [G1, G2]. However, the set of constant functions of norm at most one in L_{E}^{1} already shows that (for a reflexive infinitedimensional Banach space E), in the Bochner-Lebesgue space L_{E}^{1}, more care is needed in order to pass from weak to strong compactness. In Section 2, we extend from L_{ℝ}^{1} to L_{E}^{1} the above weak-to-norm result, along with the sequential analogue. In Section 3, limited convergence (a weakening of strong convergence [B1,B2]) is examined. Limited convergence provides an extension of the Lebesgue Dominated Convergence Theorem to L_{E}^{1}. Necessary and sufficient conditions to pass from weak to limited convergence are given. In Section 4, the concept of tightness helps to extend the results from the previous two sections. In Section 5, convergence in the Pettis norm, a weakening of strong convergence along lines distinct from limited convergence, is examined. Similarly, necessary and sufficient conditions to

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 28A20, 28A99, 46E30.

∗ Supported in part by NSF grant DMS-9204301 and DMS-9306460.

∗∗ Part of this paper is contained in the doctoral dissertation of this author, who is grateful for the guidance of his advisor Professor C. Castaing.

pass from weak to Pettis-norm convergence are given. In the study, implications between several modes of convergence on L_{E}^{1} are examined.

Throughout this paper (E, ∥ · ∥) is a Banach space with dual E^{∗} and B_{E} is the closed unit ball of E. The triple (Ω, F, μ) is a finite measure space. Without loss, we take μ to be a probability measure. For B ∈ F, we often examine the collection F^{+}(B) of all measurable subsets of B with (strictly) positive measure and denote F^{+}(Ω) by just F^{+}. By L_{E}^{1} we denote the (prequotient) space of all Bochner μ-integrable functions from Ω into E. On this space the classical L_{E}^{1}- seminorm is given by ∥f∥_{1} := ∫_{Ω} ∥f∥dμ and convergence in this seminorm is called strong convergence.

Recall [IT] that the dual of (L_{E}^{1}, ∥ · ∥_{1}) is the (prequotient) space L_{E^{∗}}^{∞}[E] of scalarly measurable bounded functions from Ω into E^{∗}. The subspace L_{E^{∗}}^{∞} of L_{E^{∗}}^{∞}[E] consisting of the strongly measurable functions actually coincides with L_{E^{∗}}^{∞}[E] if and only if E^{∗} has the Radon Nikodym property (RNP); cf. [DU, IT]. Convergence in the corresponding weak topology σ(L_{E}^{1}, L_{E^{∗}}^{∞}[E]) is called weak convergence. We will also consider the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology on L_{E}^{1}.

Also recall that a subset K of L_{E}^{1} functions is uniformly integrable if

lim_{c→∞} sup_{f∈K} ∫_{[||f||≥c]} ||f|| dμ = 0 .

It is well known [N] that K is uniformly integrable if and only if it is bounded (i.e. sup_{f∈K} ||f||_{L_{E}^{1}} is finite) and equi-integrable, i.e.

lim_{μ(A)→0^{+}} sup_{f∈K} ∫_{A} ||f|| dμ = 0 .

All notations and terminology, not otherwise explained, are as in [DU, IT, or N].

2. Weak vs. Strong Convergence in L_{E}^{1}

Our goal is to determine precisely when (via an oscillation condition) a weakly convergent sequence is also strongly convergent, along with the nonsequential analogue.

For f ∈ L_{E}^{1} and A ∈ F, the average value and the Bocce oscillation of f over A are (respectively)

m_{A}(f) := \frac{∫_{A} f dμ}{μ(A)}
Bocce-osc f|_{A} := \frac{∫_{A} ||f − m_{A}(f)|| dμ}{μ(A)}

Weak and Strong L_{E}^{1}-convergence 3

observing the convention that 0/0 is 0. The following elementary inequalities are useful

|Bocce-osc f|_{A} − Bocce-osc g|_{A}| ≤ Bocce-osc (f − g)|_{A}
Bocce-osc (f + g)|_{A} ≤ Bocce-osc f|_{A} + Bocce-osc g|_{A}
(2.1)  μ(A) Bocce-osc f|_{A} ≤ 2 ∫_{A} ||f|| dμ .

In the spirit of [G1], we consider the following oscillation conditions.

Definition 2.1. [B3] A sequence (f_{k})_{k=1}^{∞} of functions in L_{E}^{1} satisfies the sequential Bocce criterion if for each subsequence (f_{k_{j}}) of (f_{k}), each ϵ > 0, and each B in F^{+} there is a set A in F^{+}(B) such that lim inf_{j} Bocce-osc f_{k_{j}} |_{A} < ϵ .

Definition 2.2. [G1] A subset K of L_{E}^{1} satisfies the Bocce criterion if for each ϵ > 0 and each B in F^{+} there is a finite collection A of sets in F^{+}(B) such that for each f in K there is a set A in A satisfying Bocce-osc f|_{A} < ϵ .

It is known [G1, G2] that a relatively weakly compact subset of L_{ℝ}^{1} is relatively norm compact if and only if it satisfies the Bocce criterion. We now extend to L_{E}^{1}.

Theorem 2.3. A sequence (f_{k}) in L_{E}^{1} converges strongly to f_{0} in L_{E}^{1} if and only if

(1) (f_{k}) converges weakly to f_{0} in L_{E}^{1}

(2) (f_{k}) satisfies the sequential Bocce criterion

(3) Δ_{B} := {m_{B}(f_{k}) : k ∈ N} is relatively norm compact in E for each B ∈ F^{+}. Condition (1) may be replaced with

(1’) (f_{k}) converges to f_{0} in L_{E}^{1} in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology. Also, condition (3) may be replaced with

(3’) lim_{k} ||m_{B}(f_{k}) − m_{B}(f_{0})|| = 0 for each B ∈ F^{+}.

Note that Theorem 2.3 need not hold if one replaces condition (1) (resp. (1’) ) with (f_{k}) is Cauchy in the weak (resp. σ(L_{E}^{1}, L_{E^{∗}}^{∞})-) topology since L_{E}^{1} need not be sequentially complete in this topology. Recall that L_{E}^{1} is weakly sequentially complete if and only if E is [T]; on the other hand, L_{E}^{1} is σ(L_{E}^{1}, L_{E^{∗}}^{∞})-sequentially complete if and only if E is weakly sequentially complete and has the RNP (cf. [BH1], [SW]).

There is a set-analogue of Theorem 2.3:

Theorem 2.4. A subset K of L_{E}^{1} is relatively norm compact if and only if (1) K is relatively weakly compact

(2) K satisfies the Bocce criterion

(3) Δ_{B} := {m_{B}(f) : f ∈ K} is relatively norm compact in E for each B ∈ F^{+}.

Condition (1) may be replaced with (1’) K is relatively compact in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology.

Note that the above condition (3) is indispensable, as shown by Example 3.2 to come. In general, if (f_{k}) is weakly convergent (resp. K is relatively weakly compact), then the corresponding sets Δ_{B} are relatively weakly compact in E. Thus if E is finite-dimensional, then condition (3) in the above theorems is not necessary.

It is possible to prove Theorems 2.3 and 2.4 by using methods similar to those in [G2]. Here ideas from both [B3] and [G2] are combined. The following elementary lemmas are useful.

Lemma 2.5. If f is in L_{E}^{1}, then for each ϵ > 0 and B ∈ F^{+} there is a set A in F^{+}(B) such that Bocce-osc f|_{A_{0}} < ϵ for each subset A_{0} of A.

Proof. By strong measurability of f in L_{E}^{1} and Egorov’s Theorem, there exists a sequence of simple functions converging almost uniformly to f. In combination with (2.1), the remainder of the proof is clear. □

Lemma 2.6. Let ϕ: Ω → [0, +∞] be measurable. If for each ϵ > 0 and each B in F^{+} there exists a set A in F^{+}(B) such that m_{A}(ϕ) < ϵ, then ϕ(ω) = 0 for a.e. ω.

Proof. Fix ϵ > 0. Let B be the set of all ω ∈ Ω with ϕ(ω) ≥ 2ϵ. If B ∈ F^{+}, then for the corresponding set A in F^{+}(B) we would have 2ϵμ(A) < ϵμ(A), which cannot be. So B must be a null set. □

Proof of Theorem 2.3. Consider a sequence (f_{k}) in L_{E}^{1} which converges strongly to f_{0}. Conditions (1) and (3) follow immediately. Also, by (2.1) one has that

μ(A)|Bocce-osc f_{k}|_{A} − Bocce-osc f_{0}|_{A}| ≤ 2 ∫_{Ω} ∥f_{k} − f_{0}∥ dμ → 0

for each A in F^{+}. By Lemma 2.5 the singleton {f_{0}} satisfies the Bocce criterion. Thus condition (2) also holds.

As for sufficiency of (1), (2), and (3), note that to prove strong convergence it is enough to show that any subsequence (f_{n}) of (f_{k}) contains a further subsequence which converges strongly to f_{0}. By condition (1) the set (f_{k}) is uniformly integrable; hence (∥f_{k} − f_{0}∥) must also be uniformly integrable. So the subsequence (f_{n}) contains a further subsequence (f_{n_{j}}) such that (∥f_{n_{j}} − f_{0}∥) converges weakly to some (nonnegative) function ϕ in L_{ℝ}^{1}. We shall show that Lemma 2.6 applies to ϕ; this then gives ϕ = 0 a.e., which finishes the proof. To show that the lemma applies, first note that by Lemma 2.5 (applied to f_{0}) and the given Bocce property (2), the sequence (f_{k} −f_{0}) also satisfies the sequential Bocce criterion. Now fix ϵ > 0 and B in F^{+}. Let A in F^{+}(B) be as in Definition 2.1 applied to the subsequence (f_{n_{j}} −f_{0}) of (f_{k} − f_{0}), thus

lim inf_{j} Bocce-osc (f_{n_{j}} − f_{0})|_{A} < ϵ.

But by the triangle inequality

Bocce-osc (f_{n_{j}} − f_{0})|_{A} ≥ \frac{1}{μ(A)} ∫_{A} [∥f_{n_{j}} − f_{0}∥ − ∥m_{A}(f_{n_{j}} − f_{0})∥] dμ ,

so by weak convergence of (∥f_{n_{j}} − f_{0}∥) to ϕ and by the given property (3), this leads us to m_{A}(ϕ) < ϵ, which is precisely what is needed to apply Lemma 2.6. □

A close look at the proof reveals that the conditions may be slightly weakened. Using terminology and results to come in Section 3, note that condition (1) may be replaced with the two conditions that (f_{k}) is uniformly integrable and that (f_{k}) converges scalarly weakly (see Definition 3.3) to f_{0} in L_{E}^{1}. These two conditions are equivalent to (1’), as noted in Remark 3.7. Also, condition (3’) is equivalent to the two conditions that (f_{k}) converges scalarly weakly to f_{0} and condition (3). Thus, under condition (1) or (1’), condition (3) is equivalent to (3’).

Proof of Theorem 2.4. It is well-known and easy to check that a subset K of L_{E}^{1} is relatively strongly compact if and only if it satisfies condition (3) and for each η > 0 there is a finite measurable partition π of Ω such that ∫_{Ω} ∥f − E_{π}(f)∥dμ < η for each f in K. Here E_{π}(f) denotes the conditional expectation of f relative to the finite algebra generated by π.

Consider a relatively strongly compact subset K of L_{E}^{1}. Clearly conditions (1) and (3) are satisfied. To see that condition (2) holds, fix ϵ > 0 and B ∈ F^{+}. Next, from the above observation, find the partition π := {A_{1}, ⋯, A_{N}} corresponding to η := ϵμ(B). Put A = {A_{i} ∩ B ∈ F^{+} : A_{i} ∈ π}. Fix f in K. Since

∑_{i} μ(A_{i} ∩ B)Bocce-osc f|_{A_{i}∩B} ≤ ∑_{i} μ(A_{i})Bocce-osc f|_{A_{i}}

= ∫_{Ω} ∥f − ∑_{i} m_{A_{i}}(f)1_{A_{i}}∥dμ < ϵμ(B) ,

for at least one A_{i} ∩ B ∈ A we have Bocce-osc f|_{A_{i}∩B} < ϵ.

As for the sufficiency of (1), (2), and (3), note that it is enough to show relative strong sequential compactness of K. So consider a sequence (f_{k}) in K. By condition (1), there is a subsequence (f_{k_{j}}) of (f_{k}) that converges weakly to some function f_{0} in L_{E}^{1} while condition (2) implies that (f_{k_{j}}) satisfies the sequential Bocce criterion. Now an appeal to Theorem 2.3 shows that (f_{k_{j}}) converges strongly, as needed.

As for replacing (1) with (1’), recall [BH2] that for the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology, relatively compact sets and relatively sequentially compact sets coincide. □

Section 5 gives several variations of the Bocce criterion which also provide necessary and sufficient conditions to pass from weak to strong convergence (resp. compactness).

3. Limited Convergence

This section examines limited convergence, a weakening of strong convergence [B1]. Limited convergence provides an extension to L_{E}^{1} of the Vitali Convergence Theorem (VCT), thus also of the Lebesgue Dominated Convergence Theorem (LDCT). Furthermore, it extends the previous section’s results. In the next section, a tightness condition ties together limited and strong convergence and thus extends the results of this section.

Let G be the collection of all functions g: Ω × E → ℝ satisfying

(i) g(ω, 0) = 0 for each ω in Ω (ii) g(ω, ·) is weakly σ(E, E^{∗})-continuous for each ω in Ω (iii) |g(ω, ·)| ≤ C∥ · ∥ + ϕ(ω) for each ω in Ω, for some C > 0 and ϕ in L_{ℝ}^{1} (iv) g(·, f(·)) is F-measurable for each f in L_{E}^{1}.

An example of such a function g in G is given by g(ω, x) = ∑_{i=1}^{n} |x_{i}^{∗}(x)| 1_{A_{i}}(ω) where x_{i}^{∗} ∈ E^{∗} and A_{i} ∈ F. The function g given by g(ω, x) = ||x|| is in G if E is finite-dimensional (for only then does g satisfies (ii)). The class G serves as a “test class” for limited convergence (see Remark 3.9).

Definition 3.1. A sequence (f_{k}) of functions in L_{E}^{1} converges limitedly to f_{0} in L_{E}^{1} if lim_{k} ∫_{Ω} g(ω, f_{k}(ω) − f_{0}(ω)) dμ(ω) = 0 for each g ∈ G.

Strong convergence implies limited convergence. For first note that a sequence converges limitedly to f if each subsequence has a further subsequence which converges limitedly to f. Next note that a strongly convergent sequence has the property that each subsequence has a further subsequence which is pointwise a.e. strongly convergent. Lastly note that any uniformly integrable sequence (f_{k}) which is a.e. weakly null (i.e. there is a set A of full measure such that if x^{∗} ∈ E^{∗} and ω ∈ A then x^{∗}f_{k}(ω) converges to zero) converges limitedly. To see this, fix g ∈ G and put h_{k}(ω) = g(ω, f_{k}(ω)). Condition (iii) gives that the set (h_{k}) is uniformly integrable. Conditions (i) and (ii) give that (h_{k}) is a.e.-convergent to 0. So (h_{k}) converges strongly to zero and so (f_{k}) converges limitedly.

If E is finite-dimensional then strong and limited convergence coincide (consider g ∈ G given by g(ω, x) = ||x||). However, as seen by modifying the next example, for any infinite-dimensional reflexive space E there is a sequence of L_{E}^{1} functions which converges limitedly but not strongly.

Example 3.2 (limited ⇏ strong). Take (Ω, F, μ) to be the interval [0, 1], equipped with the Lebesgue σ-algebra and measure and E := ℓ^{2}. Setting f_{k} identically equal to the k-th unit vector e_{k} in ℓ^{2} gives a sequence (f_{k}) which converges limitedly but not strongly to the null function.

Limited convergence implies weak convergence since for each b ∈ L_{E^{∗}}^{∞}[E] the function g defined by g(ω, x) = ⟨x, b(ω)⟩ is in G. As for the converse implication, even for finite-dimensional E weak convergence does not imply limited convergence.

Towards a variant of the VCT–LDCT for a sequence (f_{k}) in L_{E}^{1}, we examine the corresponding sequences (x^{∗}(f_{k})) in L_{ℝ}^{1} for x^{∗} in E^{∗}.

Definition 3.3. A sequence (f_{k}) of functions in L_{E}^{1} converges scalarly strongly (resp. scalarly in measure, scalarly weakly ) to f_{0} in L_{E}^{1} if the corresponding sequence (x^{∗}(f_{k})) in L_{ℝ}^{1} converges strongly (resp. in measure, weakly ) to x^{∗}(f_{0}) for each x^{∗} in E^{∗}.

Note the following chain of strict implications:

(3.1)  strong ⇒ limited ⇒ scalarly strong ⇒ scalarly in measure .

Since for x^{∗} ∈ E^{∗} functions of the form g(ω, ·) = |x^{∗}(·)| 1_{Ω}(ω) are in G, limited convergence implies scalarly strong convergence. The other implications in (3.1) are clear.

Furthermore, the implications are strict. Example 3.2 showed the first one is not reversible. The last implication is not reversible even for E = ℝ. The next example shows that the second implication is also strict.

Example 3.4 (scalarly strong ⇏ limited). Take (Ω, F, μ), E, and (e_{k}) as in Example 3.2. Let I_{i}^{j} be the dyadic interval [(i − 1)2^{−j}, i2^{−j}) for j ∈ ℕ and i = 1, ⋯, 2^{j}. Consider the sequence (f_{k}) of the functions f_{k}: [0, 1] → ℓ^{2} given by f_{k}(ω) = 1_{I_{1}^{k}}(ω)2^{k}e_{k}. Since for every y^{∗} := (y^{j})_{j} in E^{∗} ≈ ℓ^{2}

∫_{Ω} |y^{∗} (f_{k} (ω)) | dμ(ω) = |y^{k}| ,

(f_{k}) converges scalarly strong to the null function. But for the test function g (ω, (x_{j})) = ∑_{j=1}^{∞} x_{j} 1_{I_{2}^{j+1}}(ω) in G

∫_{Ω} g (ω, f_{k} (ω)) dμ(ω) = ∫_{I_{1}^{k}} 2^{k} 1_{I_{2}^{k+1}} dμ = \frac{1}{2} .

So (f_{k}) does not converge limitedly to the null function.

Note that a scalarly strongly convergent sequence need not be uniformly integrable (as Example 3.4 shows). However, a limitedly convergent sequence, being also weakly convergent, is necessarily uniformly integrable.

Limited convergence provides the following extension of the VCT–LDCT to L_{E}^{1}.

Theorem 3.5. Let E^{∗} have the RNP. If a uniformly integrable sequence (f_{k}) converges scalarly in measure to f_{0} in L_{E}^{1}, then it also converges limitedly to f_{0}.

The necessity of the uniform integrability condition has already been noted while the necessity of E^{∗} having the RNP follows from Remark 5.4. The proof of Theorem 3.5 uses the following lemma.

Lemma 3.6. A uniformly integrable sequence (f_{k}) of L_{E}^{1} functions converges limitedly to the null function provided that, for each N ∈ ℕ, the sequence (f_{k}^{N})_{k} of truncated functions converges limitedly to the null function, where f_{k}^{N} := f_{k} 1_{[||f_{k}||≤N]}.

Proof. Fix g ∈ G with |g(ω, ·)| ≤ C∥ · ∥ + ϕ(ω). Now

| ∫_{Ω} (g (ω, f_{k} (ω)) − g (ω, f_{k}^{N} (ω))) dμ | = | ∫_{[||f_{k}||>N]} g(ω, f_{k}(ω)) dμ |
≤ C ∫_{[||f_{k}||>N]}||f_{k}|| dμ + ∫_{[||f_{k}||>N]} ϕ dμ ,

so by uniform integrability of (f_{k}) it follows that

lim_{N→∞} sup_{k} | ∫_{Ω} (g (ω, f_{k} (ω)) − g (ω, f_{k}^{N} (ω))) dμ | = 0 .

The lemma now follows with ease. □

Proof of Theorem 3.5. Without loss of generality, we assume that f_{0} = 0 and (using the previous lemma) that the f_{k}’s are uniformly bounded. Note that we may also assume that E^{∗} is separable. Indeed, by the Pettis measurability theorem [DU, Theorem II.1.2], there is a separable subspace E_{0} of E such that the f_{k}’s are essentially valued in E_{0}. Because E^{∗} has the RNP, E_{0}^{∗} must be separable [DU, Corollary VII.2.8]. Moreover, if (f_{k}) converges limitedly to 0 in L_{E_{0}}^{1}, then it also does so in L_{E}^{1}.

As noted earlier, it is enough to show that every subsequence of (f_{k}) has a further subsequence that is a.e. weakly null. We assume (w.l.o.g.) that this former subsequence is actually the entire sequence (f_{k}).

Now let (x_{i}^{∗}) be a countable dense subset of E^{∗}. For each i the sequence (x_{i}^{∗}(f_{k})) converges in measure to zero. So there exists a subsequence (f_{k_{j}}) such that for a.e. ω

lim_{j} x_{i}^{∗}(f_{k_{j}}(ω)) = 0 .

By a Cantor diagonalization argument there is a set A of full measure and a subsequence (f_{k_{p}}) such that lim_{p} x_{i}^{∗}(f_{k_{p}}(ω)) = 0 for each fixed i and each ω in A. Since the f_{k}’s are uniformly bounded and (x_{i}^{∗}) are dense in E^{∗}, this pointwise limit property extends so that lim_{p} x^{∗}(f_{k_{p}}(ω)) = 0 for each fixed x^{∗} in E^{∗} and each ω in A. Thus, (f_{k_{p}}) is a.e. weakly null, as needed. □

Limited convergence also provides an extension of the results from the previous section; namely, it is possible to pass from weak to limited convergence via an oscillation condition. The following string of strict implications summarizes the ideas thus far.

scalarly strong ⇒ scalarly weak ⇐ σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology ⇐ weak .

Remark 3.7. A scalarly weakly convergent sequence converges in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})- topology if and only if it is uniformly integrable. (Recall that the simple functions are not dense in L_{E^{∗}}^{∞} for infinite-dimensional E.) Convergence in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})- topology implies weak convergence if and only if E^{∗} has the RNP [cf. DG] .

In the light of these observations and Theorem 3.5, we have the following variant of Theorem 2.3 for limited convergence.

Theorem 3.8. Let E^{∗} have the RNP. A sequence (f_{k}) of L_{E}^{1} functions converges limitedly to f_{0} in L_{E}^{1} if and only if

(1) (f_{k}) converges weakly to f_{0} in L_{E}^{1}

(2) (x^{∗}(f_{k})) satisfies the sequential Bocce criterion for each x^{∗} in E^{∗}. Condition (1) may be replaced with

(1’) (f_{k}) converges to f_{0} in L_{E}^{1} in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology.

Remark 3.9. Limited convergence for separable reflexive E was introduced in [B1, B2]. There, the condition (iv) is replaced with

(iv’) g is F ⊗ B(E)-measurable, Of course (iv’) always implies (iv). To see that (iv) implies (iv’) if E is separable, consider a function g which satisfies (iv). For each k ∈ ℕ, write 1_{E} = ∑_{n} 1_{E_{n}^{k}} where E_{n}^{k} ∈ B(E) and the diameter of E_{n}^{k} is less than \frac{1}{k}. Choose x_{n}^{k} ∈ E_{n}^{k}. Define g_{k}: Ω × E → ℝ by

g_{k}(ω, x) = ∑_{n} g(ω, x_{n}^{k}) 1_{E_{n}^{k}}(x) .

Since each g_{k} is F × B(E)-measurable and g_{k} converges to g almost everywhere, g is also F ⊗ B(E)-measurable.

4. The Tightness Connection

The concept of tightness links strong and limited convergence. In this section, we assume that E is a separable Banach space. Tightness is considered here with respect to the norm topology on E and only for functions. The following formulation of tightness is given in [B4].

Definition 4.1. A subset L of L_{E}^{1} is tight if there exists an F ⊗ B(E)-measurable function h: Ω × E → [0, +∞] such that

sup_{f∈L} ∫_{Ω} h(ω, f(ω)) dμ(ω) < +∞

and such that {x ∈ E : h(ω, x) ≤ β} is compact for each ω ∈ Ω and each β ∈ ℝ.

In [Jaw], the following equivalent formulation of tightness is observed.

Definition 4.1′. A subset L of L_{E}^{1} is tight if for each ϵ > 0 there exists a measurable multifunction F_{ϵ} from Ω to the compact subsets of E such that

μ ({ω ∈ Ω: f(ω) ∉ F_{ϵ}(ω)}) ≤ ϵ

for each f ∈ L. We say that such a multifunction F_{ϵ} is measurable (i.e. graphmeasurable) if its graph {(ω, x) ∈ Ω × E : x ∈ F_{ϵ}(ω)} is an F ⊗ B(E)-measurable subset of Ω × E.

To see the equivalence in one direction, denote the supremum in Definition 4.1 by σ and define F_{ϵ}(ω) as the set of all x ∈ E for which h(ω, x) ≤ σ/ϵ. In the other direction, one obtains a sequence (F_{n}) of compact-valued multifunctions by letting F_{n} correspond to ϵ = 3^{−n} in Definition 4.1′. Without loss of generality we may suppose that (F_{n}(ω)) is nondecreasing (rather than taking finite unions ∪_{m≤n}F_{m}). Now a function h satisfying the requirements of Definition 4.1 is obtained by setting h(ω, x) := 2^{n} for x ∈ F_{n}(ω)\F_{n−1}(ω) with F_{0}(ω) := ∅ and h(ω, x) := +∞ for x ∈ E \ ∪_{n}F_{n}(ω).

In Definition 4.1′ we may assume without loss of generality that F_{ϵ}(ω) is convex and contains 0 for each ω in Ω by consider the corresponding multifunction ω ⟼ \overline{co}(F_{ϵ}(ω) ∪ \{0\}). The measurability of this new map follows from [CV, Theorem III.40] and [HU, Remark (1), p. 163]. Therefore, if L is tight and (B_{f})_{f∈L} is a family of sets from F, then the set {f 1_{B_{f}} : f ∈ L} is also tight. Note that a bounded sequence in L_{E}^{1} is tight if E is finite dimensional (simply take h(ω, x) := ||x|| in Definition 4.1). For further details on tightness see [B4, B5].

Recall the following fact [ACV, Théorème 6].

Fact 4.2. A uniformly integrable tight subset of L_{E}^{1} is relatively weakly compact.

Although weak compactness is not sufficient to guarantee that the corresponding subset Δ_{B} are relatively norm compact (consider Example 3.2), the following generalization of a result of Castaing [C1] shows that uniform integrability plus tightness is sufficient.

Lemma 4.3. Let L be a tight uniformly integrable subset of L_{E}^{1}. Then Δ_{B} := {m_{B}(f) : f ∈ L} is relatively norm compact in E for each B in F^{+}.

Proof. Let the subset L of L_{E}^{1} be uniformly integrable and tight. Since for each B ∈ F^{+} the set {f 1_{B}: f ∈ L} is also uniformly integrable and tight, it is enough to show that Δ_{Ω} is relatively norm compact. Arguing as in Remark (1) on p. 163 of [HU], we may suppose without loss of generality that F is complete.

Fix δ > 0. By the uniform integrability of L, there exist α > 0 and ϵ > 0 such that for each set A of measure at most ϵ we have that

sup_{f∈L} ∫_{A} ||f|| dμ ≤ δ/2 and sup_{f∈L} ∫_{[||f||>α]} ||f|| dμ ≤ δ/2 .

Let F_{ϵ} be a multifunction given by Definition 4.1′ and G_{ϵ}^{α} = F_{ϵ}∩αB_{E} (i.e. G_{ϵ}^{α}(ω) = F_{ϵ}(ω)∩αB_{E}, ∀ω ∈ Ω). Since G_{ϵ}^{α} is convex compact valued and integrably bounded (that means ||G_{ϵ}^{α}|| = sup{||x||: x ∈ G_{ϵ}^{α}(ω)} ∈ L_{ℝ_{+}}^{1}), the subset K_{ϵ}^{α} = {∫_{Ω} G_{ϵ}^{α} dμ} is convex and compact in E [CV, Theorem V.15]. Let now A_{f}^{ϵ} be the set of all ω ∈ Ω with f(ω) ∈ F_{ϵ}(ω). Note that μ(Ω \ A_{f}^{ϵ}) ≤ ϵ. Since for each f ∈ L

∫_{[||f||≤α]} f 1_{A_{f}^{ϵ}} dμ ∈ K_{ϵ}^{α} ,

the set Δ_{Ω}^{ϵ,α} := {∫_{[||f||≤α]} f 1_{A_{f}^{ϵ}} dμ: f ∈ L} is relatively compact in E. Moreover, the distance between Δ_{Ω}^{ϵ,α} and Δ_{Ω} is at most δ since

|| ∫_{Ω} f dμ − ∫_{[||f||≤α]} f 1_{A_{f}^{ϵ}} dμ || ≤ ∫_{[||f||>α]} ||f|| dμ + ∫_{Ω} ||f 1_{Ω\A_{f}^{ϵ}} || dμ ≤ δ

for each f ∈ L. Thus Δ_{Ω} is relatively compact. □

Measure convergent sequences enjoy tightness.

Lemma 4.4. A sequence in L_{E}^{1} which converges in measure is tight.

Proof. Consider a sequence (f_{k}) in L_{E}^{1} which converges in measure to f_{0}. For each natural number k, let λ_{k} be the bounded non-negative image measure on E induced by μ and the measurable function f_{k}: Ω → E. Since E is a Radon space (thanks to the separability assumption), λ_{k} is a Radon (or tight) measure. For each bounded continuous function ϕ ∈ C^{b}(E), we have

(4.1)  λ_{k}(ϕ) = ∫_{Ω} ϕ(f_{k}(ω)) dμ(ω) .

It is easy to see that the measure convergence of (f_{k}) to f_{0} in L_{E}^{1} implies the narrow convergence (or weak convergence in the σ(M^{b}(E), C^{b}(E))-topology) of (λ_{k}) to λ_{0}. For otherwise there would exist ϕ ∈ C^{b}(E) and a subsequence (f_{k_{j}}) converging almost everywhere to f_{0} and such that (λ_{k_{j}}(ϕ)) does not converge to λ_{0}(ϕ). But by (4.1), this contradicts the Lebesgue Dominated Convergence Theorem. Therefore, the sequence (λ_{k}) is tight in M^{b}(E) in the classical sense [S, Appendix Theorem 4], which implies that (f_{k}) is tight in the sense of Definitions 4.1 and 4.1′. □

From Lemmas 4.3 and 4.4, the following reformulation of Theorem 2.3 follows with ease.

Theorem 4.5. A sequence (f_{k}) in L_{E}^{1} converges strongly to f_{0} in L_{E}^{1} if and only if

(1) (f_{k}) converges weakly to f_{0} in L_{E}^{1}

(2) (f_{k}) satisfies the sequential Bocce criterion

(3) (f_{k}) is tight.

Tightness connects strong and limited convergence.

Theorem 4.6. A sequence (f_{k}) of L_{E}^{1} converges strongly to f_{0} if and only if (f_{k}) is tight and converges limitedly to f_{0}.

Before proceeding with the proof of Theorem 4.6, we note some immediate corollaries.

Theorem 3.5 - revisited. Let E^{∗} have the RNP and E be separable. If a uniformly integrable tight sequence (f_{k}) converges scalarly in measure to f_{0} in L_{E}^{1}, then it also converges strongly to f_{0}.

Theorem 3.8 - revisited. Let E^{∗} have the RNP and E be separable. A sequence (f_{k}) of L_{E}^{1} functions converges strongly to f_{0} in L_{E}^{1} if and only if

(1) (f_{k}) converges weakly to f_{0} in L_{E}^{1}

(2) (x^{∗}(f_{k})) satisfies the sequential Bocce criterion for each x^{∗} in E^{∗}

(3) (f_{k}) is tight. Condition (1) may be replaced with

(1’) (f_{k}) converges to f_{0} in L_{E}^{1} in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology.

The proof of Theorem 4.6 uses the following standard fact (compare with Lemma 3.6).

Fact 4.7. A uniformly integrable sequence (f_{k}) of L_{E}^{1} functions converges strongly to the null function provided that, for each N ∈ ℕ, the sequence (f_{k}^{N})_{k} of truncated functions converges strongly to the null function, where f_{k}^{N} := f_{k} 1_{[||f_{k}||≤N]}.

Proof of Theorem 4.6. The implication in one direction follows from our previous work. As for the other direction, let (f_{k}) be a tight sequence in L_{E}^{1} which converges limitedly to f_{0}. Because the image measure of μ under f_{0} is a Radon measure on E, the singleton {f_{0}} must be tight. Since the union of two tight sets is again tight, we have that the set {f_{k} : k ∈ ℕ ∪ {0}} is also tight; let h correspond to this set as in Definition 4.1. Without loss of generality, we assume that f_{0} is the null function and that the f_{k}’s are uniformly bounded (in L_{E}^{∞}) by some M > 0. To avoid the non-metrizability of the σ(E, E^{∗})-topology, we use ideas from [B5]. By well-known facts about Suslin spaces [S, Corollary 2 of Theorem II.10], there exists a metric d on E defining a topology τ_{d} weaker than the weak topology σ(E, E^{∗}) and such that (E, τ_{d}) is a Suslin space. Define ϕ: Ω × E → ℝ by ϕ(ω, x) := max(−∥x∥, −M). For each ϵ > 0, consider the function ϕ_{ϵ} : Ω × E → ℝ given by

ϕ_{ϵ}(ω, x) := ϕ(ω, x) + ϵh(ω, x).

From the inf-compactness property of h (see Definition 4.1) it follows that ϕ_{ϵ}(ω, ·) is also inf-compact on E for each ω ∈ Ω and ϵ > 0; in turn, this implies infcompactness of the same functions for the weak topology σ(E, E^{∗}) and hence for τ_{d}. Moreover, the F ⊗ B(E)-measurability^{1} of ϕ_{ϵ} is evident.

For each ϵ > 0 and p ∈ ℕ we define the approximate function ϕ_{ϵ}^{p}: Ω × E → ℝ by

ϕ_{ϵ}^{p}(ω, x) = inf_{y∈E}{ϕ_{ϵ}(ω, y) + p d(x, y)}.

Evidently, for each ϵ > 0 the sequence (ϕ_{ϵ}^{p}) is (pointwise) nondecreasing. It is well-known [B4,V1] that ϕ_{ϵ}^{p} has an F ⊗ B(E)-measurable modification ψ_{ϵ}^{p} (i.e., ψ_{ϵ}^{p}(ω, ·) = ϕ_{ϵ}^{p}(ω, ·) a.e.) such that for each ω ∈ Ω the function ψ_{ϵ}^{p}(ω, ·) is d-Lipschitz continuous on E and therefore is σ(E, E^{∗})-continuous. Furthermore, as a wellknown property of this approximation, by τ_{d}-lower semicontinuity and boundedness below of ϕ_{ϵ}(ω, ·), we have

ϕ_{ϵ}(ω, x) = lim_{p} ↑ ψ_{ϵ}^{p}(ω, x)

for a.e. ω and each x ∈ E. We now set ψ̂_{ϵ}^{p}(ω, x) = min(ψ_{ϵ}^{p}(ω, x) − ψ_{ϵ}^{p}(ω, 0), p). Note that −M − ϵh(ω, 0) ≤ ψ̂_{ϵ}^{p}(ω, ·) ≤ p for a.e. ω, where ω ↦ h(ω, 0) is integrable in view of Definition 4.1 and f_{0} ≡ 0. For each ϵ > 0 and p ∈ ℕ, the function ψ̂_{ϵ}^{p} satisfies the conditions (i) to (iv) for the test functions of G. Therefore, by the limited convergence of the f_{k}’s, we have

lim_{k→∞} ∫_{Ω} ψ̂_{ϵ}^{p}(ω, f_{k}(ω)) dμ(ω) = 0.

It follows that

lim inf_{k→∞} ∫_{Ω} ψ_{ϵ}^{p}(ω, f_{k}(ω)) dμ(ω) ≥ ∫_{Ω} ψ_{ϵ}^{p}(ω, 0) dμ(ω) .

Thus, for each ϵ > 0 and p ∈ ℕ

lim inf_{k→∞} ∫_{Ω} ϕ_{ϵ}(ω, f_{k}(ω)) dμ(ω) ≥ lim inf_{k→∞} ∫_{Ω} ψ_{ϵ}^{p}(ω, f_{k}(ω)) dμ(ω) ≥ ∫_{Ω} ψ_{ϵ}^{p}(ω, 0) dμ(ω) .

1For any of the three topologies E is a Suslin space; hence, it has the same Borel σ-algebra B(E).

The monotone convergence theorem gives, for each ϵ > 0

α_{ϵ} := lim inf_{k→∞} ∫_{Ω} ϕ_{ϵ}(ω, f_{k}(ω)) dμ(ω) ≥ lim_{p} ↑ ∫_{Ω} ψ_{ϵ}^{p}(ω, 0) dμ(ω)
= ∫_{Ω} ϕ_{ϵ}(ω, 0) dμ(ω) = ϵ ∫_{Ω} h(ω, 0)dμ(ω) ,

thus

0 ≤ α_{ϵ} ≤ lim inf_{k→∞} ∫_{Ω} ϕ(ω, f_{k}(ω)) dμ(ω) + ϵ sup_{k} ∫_{Ω} h(ω, f_{k}(ω)) dμ(ω) .

Since ϕ(ω, f_{k}(ω)) = −||f_{k}(ω)||, by our initial assumption, the proof is finished by letting ϵ go to zero. □

Fact 4.2 and Theorem 4.5 gives that a uniformly integrable tight sequence in L_{E}^{1} which satisfies the sequential Bocce criterion has a strongly convergent subsequence. Recall that a sequence (f_{k}) is said to be bounded if sup_{k} ||f_{k}||_{L_{E}^{1}} is finite. In the above, if we relax uniform integrability to boundedness, we need not have strong subsequential convergence (just consider the sequence (n 1_{[0,1/n]})_{n} in L_{ℝ}^{1}) but we do have measure subsequential convergence. We can state this result as a strong Biting lemma.

Theorem 4.8. Let (f_{k}) be a bounded tight sequence in L_{E}^{1} satisfying the sequential Bocce criterion. Then there exist a subsequence, say (f_{n}), of (f_{k}) and an increasing sequence (A_{n}) in F such that

(1) lim_{n→∞} μ(A_{n}) = μ(Ω)

(2) the sequence (f_{n} 1_{A_{n}}) converges strongly in L_{E}^{1}

(3) the sequence (f_{n} 1_{Ω\A_{n}}) converges to 0 in measure. Therefore, the subsequence (f_{n}) converges in measure.

The proof uses Gaposhkin’s Biting lemma [Ga, Lemma C], which is also referred to as Slaby’s Biting lemma [cf. C2].

Biting lemma. Let (f_{k}) be a bounded sequence in L_{E}^{1}. Then there exist a subsequence, say (f_{n}), of (f_{k}) and an increasing sequence (A_{n}) in F such that

(1) lim_{n→∞} μ(A_{n}) = μ(Ω)

(2) the sequence (f_{n} 1_{A_{n}}) is uniformly integrable in L_{E}^{1} . Note that (1) implies that the sequence (f_{n} 1_{Ω\A_{n}}) converges to 0 in measure.

Proof of Theorem 4.8. Consider a bounded tight sequence (f_{k}) in L_{E}^{1} which satisfies the sequential Bocce criterion. Apply the Biting lemma to find the corresponding subsequence, say (f_{n}), of (f_{k}) and sequence (A_{n}) in F. Since (f_{n} 1_{A_{n}}) is uniformly integrable and tight, it is relatively weakly sequentially compact. By passing to a further subsequence we can assume that (f_{n} 1_{A_{n}}) converges weakly in L_{E}^{1}. Since (f_{k}) satisfies the sequential Bocce criterion, using condition (1) it is easy to check that (f_{n} 1_{A_{n}}) also satisfies the sequential Bocce criterion (in the definition, for a fixed B ∈ F^{+}(Ω), apply the criterion to B_{0} := B ∩ A_{N} for a sufficiently large N). Theorem 4.5 gives that (f_{n} 1_{A_{n}}) converges strongly. □

5. Pettis Norm

This section examines Pettis norm convergence in light of the previous sections.

Definition 5.1. A strongly measurable function f: Ω → E is Pettis integrable if x^{∗}(f) belongs to L_{ℝ}^{1} for every x^{∗} in E^{∗} and if for every B in F there exists x_{B} in E such that

∫_{B} x^{∗}(f)dμ = x^{∗}(x_{B}) for all x^{∗} ∈ E^{∗}.

The space P_{E}^{1} of (equivalence classes of) all strongly measurable Pettis integrable functions forms a normed linear space under the Pettis (semi )norm

||f||_{Pettis} = sup_{x^{∗}∈B_{E^{∗}}} ∫_{Ω} |x^{∗}(f)| dμ .

Clearly P_{E}^{1} contains L_{E}^{1}, to which we restrict considerations.

In general, Pettis norm convergence on L_{E}^{1} is incomparable with limited convergence but is comparable with the other modes of convergence in chain (3.1). A parallel chain of strict implications is

(5.1)  strong ⇒ Pettis ⇒ scalarly strong .

Note that when E is finite-dimensional, the two chains (3.1) and (5.1) merge into

strong ⇔ Pettis ⇔ limited ⇔ scalarly strong .

The implications in chain (5.1) are clear; the following two examples show that they are strict.

Example 5.2 (scalarly strong ⇏ Pettis). Example 3.2 suffices here but, for later use, we consider the following variation. Take (Ω, F, μ), E := ℓ^{2}, and (e_{k}) as in Example 3.2. Consider the Rademacher-type functions f_{k}: [0, 1] → ℓ^{2} defined by f_{k}(ω) := e_{k}r_{k}(ω) where r_{k} is the k-th Rademacher function. Clearly, (f_{k}) converges scalarly strong to the null function yet the Pettis norm of each f_{k} is one.

Example 5.3 (Pettis ⇏ strong). [P] Take (Ω, F, μ), E, (e_{k}), and (I_{i}^{j}) as in Example 3.4. Consider the sequence (f_{k}) of the integrable functions f_{k}: Ω → ℓ^{2} given by f_{k}(ω) := ∑_{i=1}^{2^{k}} 1_{I_{i}^{k}}(ω)e_{2^{k}+i}. To see that (f_{k}) converges in the Pettis norm to the null function, fix y^{∗} := (y^{i})_{i} ∈ B_{ℓ^{2}}. Put \overline{y}^{∗} := (|y^{i}|)_{i} and note that

∫_{Ω} |y^{∗}(f_{k})| dμ = ∑_{i=1}^{2^{k}} |y^{2^{k}+i}| μ(I_{i}^{k}) = 2^{−k} \overline{y}^{∗} (∑_{i=1}^{2^{k}} e_{2^{k}+i})

≤ 2^{−k} ∥∑_{i=1}^{2^{k}} e_{2^{k}+i}∥_{ℓ^{2}} = 2^{− \frac{k}{2}} .

Thus ||f_{k}||_{Pettis} → 0. But (f_{k}) does not converge strongly since ∫_{Ω} ||f_{k}||_{ℓ^{2}} dμ = 1.

Example 5.3 illustrates (consider g_{k} := 2^{\frac{k}{4}} f_{k}) that a Pettis-norm convergent sequence need not be uniformly integrable. Example 3.2 shows that a limitedly convergent sequence need not converge in the Pettis norm. Theorem 3.5 gives that if E^{∗} has the RNP, then a uniformly integrable Pettis-norm convergent sequence in L_{E}^{1} also converges limitedly. The following remark shows the necessity of E^{∗} having the RNP.

Remark 5.4 [DG]. A uniformly integrable Pettis-norm convergent sequence also converges in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology and, if furthermore E^{∗} has the RNP, then also weakly. But if E^{∗} fails the RNP, then there is an essentially bounded sequence which converges in the Pettis norm but not weakly (thus not limitedly).

In the case that E = ℓ^{1}, this sequence is easy to construct.

Example 5.5 (Pettis ⇏ limited). Let (Ω, F, μ) be as in Example 3.2 and let E = ℓ^{1}. Consider the sequence (f_{k}) in L_{E}^{1} given by f_{k}(ω) := \frac{1}{k} ∑_{i=1}^{k} r_{i}(ω)e_{i}, where e_{i} is the i-th unit vector in ℓ^{1} and r_{i} is the i-th Rademacher function. Note that (f_{k}) is essentially bounded. As for the Pettis norm of f_{k}, fix y^{∗} = (y^{i})_{i} ∈ E^{∗} = ℓ^{∞}. Since

∫_{Ω} |y^{∗}(f_{k})| dμ = \frac{1}{k} ∫_{Ω} |∑_{i=1}^{k} y^{i} r_{i}(ω)| μ(dω) ,

Khintchine’s inequality [cf. D1] shows that ||f_{k}||_{Pettis} behaves like \frac{1}{\sqrt{k}} and so ||f_{k}||_{Pettis} → 0. Thus (f_{k}) converges scalarly weakly to the null function and so if it also converges limitedly or weakly, it does so to the null function. But consider b ∈ L_{E^{∗}}^{∞}[E] ≈ L_{E}^{1}^{∗} given by b(ω) := (1_{[r_{i}=1]}(ω))_{i}, along with the corresponding test function g(ω, x) := ⟨x, b(ω)⟩. Since ∫_{Ω}⟨f_{k}(ω), b(ω)⟩ dμ(ω) = \frac{1}{2} we see that (f_{k}) does not converge limitedly nor weakly.

At this time there is no analogue to Theorem 3.5 which would allow one to pass from scalarly in measure convergence to Pettis-norm convergence when E^{∗} has the RNP. Note that if the sequence (f_{k}) is Cauchy in the Pettis norm, then the corresponding subsets Δ_{B} of E are relatively norm compact for each B ∈ F^{+}.

But even for an essentially bounded (thus uniformly integrable) sequence (f_{k}) for which the Δ_{B} are all relatively norm compact, the implication scalarly in measure ⇒ Pettis does not hold in general, as shown by Example 5.2.

It is possible in certain situations to pass from weak to Pettis-norm convergence. For this, a measurement of the oscillation relative to the Pettis norm is needed.

Definition 5.6. For f ∈ L_{E}^{1} and A ∈ F the Pettis Bocce oscillation of f over A is

Pettis-Bocce-osc f|_{A} := sup_{x^{∗}∈B_{E^{∗}}} Bocce-osc x^{∗}(f)|_{A} .

Since Bocce-osc x^{∗}f|_{A} is at most ∥x^{∗}∥ Bocce-osc f|_{A}, the Pettis-Bocce-osc f|_{A} is at most Bocce-osc f|_{A}.

Definition 5.7. A sequence (f_{k}) of functions in L_{E}^{1} satisfies the sequential Pettis Bocce criterion if for each subsequence (f_{k_{j}}) of (f_{k}), each ϵ > 0, and each B in F^{+}, there is a set A in F^{+}(B) such that lim inf_{j} Pettis-Bocce-osc f_{k_{j}} |_{A} < ϵ .

Definition 5.8. A subset K of L_{E}^{1} is Pettis uniformly integrable if the corresponding subset K̃ := {x^{∗}f: x^{∗} ∈ B_{E^{∗}}, f ∈ K} of L_{ℝ}^{1} is uniformly integrable.

Clearly, K is Pettis uniformly integrable if and only if it is Pettis-norm bounded and the corresponding set K̃ is equi-integrable.

The following variants of Lemma 2.5 and Lemma 2.6, respectfully, are useful.

Lemma 5.9. The sequential Pettis Bocce criterion is translation invariant.

Proof. Let the sequence (f_{k}) satisfy the sequential Pettis Bocce criterion and fix f ∈ L_{E}^{1}. The fact that (f_{k} + f) also satisfies the Pettis Bocce criterion follows directly from the definition, Lemma 2.5, and the observation that (cf. inequalities 2.1)

Pettis-Bocce-osc (f_{k} + f)|_{A} ≤ Pettis-Bocce-osc f_{k}|_{A} + Pettis-Bocce-osc f|_{A}. □

Lemma 5.10. Let (f_{k}) be a Pettis uniformly integrable sequence in L_{E}^{1}. If for each subsequence (f_{k_{j}}) of (f_{k}), each ϵ > 0, and each B in F^{+}, there exists a subset A in F^{+}(B) such that

lim inf_{j} sup_{x^{∗}∈B_{E^{∗}}} \frac{∫_{A} |x^{∗}(f_{k_{j}})|dμ}{μ(A)} < ϵ

then (f_{k}) converges to 0 in the Pettis norm.

Proof. Assume (f_{k}) is Pettis uniformly integrable but does not converge to 0 in the Pettis norm. Since (f_{k}) is Pettis uniformly integrable, the subset {|x^{∗}(f_{k})| : x^{∗} ∈ B_{E^{∗}}, k ∈ ℕ} of L_{ℝ}^{1} is relatively weakly compact. So there exists ϵ > 0, a subsequence (f_{k_{j}}) of (f_{k}), a sequence (x_{k_{j}}^{∗}) in B_{E^{∗}}, and g in L_{ℝ}^{1} such that 2ϵ < ∫_{Ω} |x_{k_{j}}^{∗}(f_{k_{j}})|dμ and |x_{k_{j}}^{∗}(f_{k_{j}})| → g weakly in L_{ℝ}^{1}. Since 2ϵ ≤ ∫_{Ω} g dμ, the set B := [g > ϵ] is in F^{+}. For any subset A of B with positive measure

lim inf_{j} sup_{x^{∗}∈B_{E^{∗}}} \frac{∫_{A} |x^{∗}(f_{k_{j}})|dμ}{μ(A)} ≥ lim inf_{j} \frac{∫_{A} |x_{k_{j}}^{∗}(f_{k_{j}})|dμ}{μ(A)} = \frac{∫_{A} g dμ}{μ(A)} > ϵ.

Thus the lemma holds. □

The Pettis-norm analogue to Theorems 2.3 and 3.8 now follows with ease.

Theorem 5.11. A sequence (f_{k}) in L_{E}^{1} converges in the Pettis norm to f_{0} in L_{E}^{1} if and only if

(1) (f_{k}) is Pettis uniformly integrable

(2) (f_{k}) satisfies the sequential Pettis Bocce criterion

(3) lim_{k} ∥m_{B}(f_{k}) − m_{B}(f_{0})∥ = 0 for each B ∈ F^{+}.

Proof. Consider a sequence (f_{k}) that converges in the Pettis norm to f_{0} in L_{E}^{1}. It is easy to check that conditions (1) and (3) hold. Since for x^{∗} in B_{E^{∗}} and A in F^{+}

|Bocce-osc x^{∗}(f_{k})|_{A} − Bocce-osc x^{∗}(f_{0})|_{A}| ≤ \frac{2}{μ(A)}∥f_{k} − f_{0}∥_{Pettis}

and Bocce-osc x^{∗}(f_{0})|_{A} ≤ Bocce-osc f_{0}|_{A}, from Lemma 2.5 we see that (f_{k}) satisfies the sequential Pettis Bocce criterion.

As for the other implication, consider a sequence (f_{k}) which satisfies conditions (1), (2) and (3). To show that f_{k} → f_{0} in the Pettis norm, we will show that (f_{k} − f_{0}) satisfies the conditions of Lemma 5.10. First note that condition (1) gives that (f_{k} − f_{0}) is Pettis uniformly integrable. Fix ϵ > 0 and B in F^{+}. Consider a subsequence (f_{k_{j}}) of (f_{k}). Since (f_{k} − f_{0}) satisfies the sequential Pettis Bocce criterion, there is a set A in F^{+}(B) such that lim inf_{j} Pettis-Bocce-osc (f_{k_{j}} −f_{0})|_{A} < ϵ. Since

sup_{x^{∗}∈B_{E^{∗}}} \frac{∫_{A} |x^{∗}(f_{k_{j}} − f_{0})|dμ}{μ(A)} − ∥m_{A}(f_{k_{j}} − f_{0})∥ ≤ Pettis-Bocce-osc (f_{k_{j}} − f_{0})|_{A},

using (3) we see that

lim inf_{j} sup_{x^{∗}∈B_{E^{∗}}} \frac{∫_{A} |x^{∗}(f_{k_{j}} − f_{0})|dμ}{μ(A)} < ϵ

as needed. Thus f_{k} → f_{0} in the Pettis norm. □

Remark 5.4 ties weak convergence into Theorem 5.11.

Corollary 5.12. A sequence (f_{k}) in L_{E}^{1} converges in the Pettis norm to f_{0} in L_{E}^{1} and is uniformly integrable if and only if

(1) (f_{k}) converges to f_{0} in the σ(L_{E}^{1}, L_{E^{∗}}^{∞})-topology

(2) (f_{k}) satisfies the sequential Pettis Bocce criterion

(3) Δ_{B} := {m_{B}(f_{k}) : k ∈ N} is relatively norm compact in E for each B in

F^{+}. Condition (3) may be replaced by

(3’) lim_{k} ||m_{B}(f_{k}) − m_{B}(f_{0})|| = 0 for each B in F^{+}. Furthermore, if E^{∗} has the RNP, then (1) is equivalent to

(1’) (f_{k}) converges to f_{0} weakly in L_{E}^{1}.

Note that under (1), conditions (3) and (3’) are equivalent.

Since a Pettis convergent sequence need not be tight (consider Example 5.5 along with Fact 4.2), there is no Pettis-analogue to Theorem 4.5.

6. Variation of the Bocce Criterion

As noted in this section, several variations of the sequential Bocce criterion also provided necessary and sufficient conditions to pass from weak to strong convergence. For a sequence (f_{k}) of functions in L_{E}^{1}, consider the following Bocce-like oscillation conditions.

The sequence (f_{k}) satisfies oscillation condition (B0) if for each ϵ > 0 and each B in F^{+} there is a set C in F^{+}(B) and N ∈ ℕ such that

Bocce-osc f_{k}|_{C} < ϵ

for each k ≥ N.

The sequence (f_{k}) satisfies oscillation condition (B1) if for each ϵ > 0 there is a finite measurable partition π = (A_{i})_{i=0}^{p} of Ω with μ(A_{0}) < ϵ and N ∈ ℕ such that

Bocce-osc f_{k}|_{A_{i}} < ϵ

for each k ≥ N and 1 ≤ i ≤ p.

The sequence (f_{k}) satisfies oscillation condition (B2) if for each ϵ > 0 there is a finite measurable partition π = (A_{i})_{i=0}^{p} of Ω with μ(A_{0}) < ϵ such that for each collection (B_{i})_{i=1}^{p} of sets with B_{i} in F^{+}(A_{i}) there is N ∈ ℕ such that

Bocce-osc f_{k}|_{B_{i}} < ϵ

for each k ≥ N and 1 ≤ i ≤ p.

The above 3 oscillation conditions have appeared in the literature [V2, B3, J] under various names. In [J], it is shown that

(B2) ⇒ (B1) ⇔ (B0) .

The proof that (B2) implies (B0) and the proof that (B1) implies (B0) are both straightforward while the proof that (B0) implies (B1) involves an exhaustion argument. It is straightforward [cf. G1] to show that (B1) implies the sequential Bocce criterion.

If the sequence (f_{k}) in L_{E}^{1} converges strongly then it satisfies (B2). This follows from minor variations of earlier arguments and noting that Lemma 2.5 may be strengthened.

Lemma 2.5 – revisited. Let f be in L_{E}^{1}. For every ϵ > 0 there is a finite measurable partition π = (A_{i})_{i=0}^{p} of Ω with μ(A_{0}) < ϵ such that for each collection (B_{i})_{i=1}^{p} of sets with B_{i} in F^{+}(A_{i})

Bocce-osc f|_{B_{i}} < ϵ

for each 1 ≤ i ≤ p.

Thus in Theorem 2.3 (and thus also in the related theorems) oscillation condition (2) may be replaced with the condition that (f_{k}) satisfies either oscillation condition (B2), (B1), or (B0).

As for the subset analogue, recall [G1] that a subset K of L_{E}^{1} is a set of small Bocce oscillation if for each ϵ > 0 there is a finite measurable partition π = (A_{i})_{i=1}^{p} of Ω such that for each f in K

∑_{i=1}^{p} μ(A_{i}) Bocce-osc f|_{A_{i}} < ϵ .

As in the L_{ℝ}^{1} case [G1], a relatively strongly compact set is a set of small Bocce oscillation and a set of small Bocce oscillation satisfies the Bocce criterion. Thus in Theorem 2.4 the oscillation condition (2) may be replaced by the condition that K be a set of small Bocce oscillation.

Acknowledgment

The authors express our appreciation to the Department of Mathematics at the University of South Carolina for providing the financial support which enabled us to start this work. We are grateful to M. Valadier and M. Moussaoui for their suggestions and discussions concerning the Section 4 and for the anonymous referee’s careful suggestions that helped clarify many fine details. B. Dawson [Da] has independently obtained results in the spirit of this paper’s Theorem 2.4.

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Mathematical Institute, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, Netherlands

E-mail address: balder@math.ruu.nl

Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.

E-mail address: girardi@math.scarolina.edu

Département de Mathématiques, Université Montpellier II, 34095 Montpellier Cedex 05, France

Current address: Département de Mathématiques, Université de Limoges, 87060 Limoges Cedex, France

E-mail address: jalby@math.univ-montp2.fr