COMPACTNESS IN L_{1}, DUNFORD-PETTIS OPERATORS, GEOMETRY OF BANACH SPACES

Maria Girardi University of Illinois at Urbana-Champaign

Proc. Amer. Math. Soc. 111 (1991) 767–777 Abstract. A type of oscillation modeled on BMO is introduced to characterize norm compactness in L_{1}. This result is used to characterize the bounded linear operators from L_{1} into a Banach space 𝔛 that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). This characterization is used to study the geometry of Banach spaces 𝔛 with the property that all bounded linear operators from L_{1} into 𝔛 are Dunford-Pettis.

1 : Introduction

The main result of this paper is a BMO-style oscillation characterization of L_{1}- norm compactness. From this result we obtain a characterization of the bounded linear operators from L_{1} into a Banach space 𝔛 that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). With such characterizations in hand, we study the geometry of Banach spaces 𝔛 with the property that all bounded linear operators from L_{1} into 𝔛 are Dunford-Pettis.

One way to insure that a relatively weakly compact set K in L_{1} is relatively norm compact is to be able to find, for each ϵ > 0, a finite measurable partition π of [0, 1] such that for each f in K and each A in π,

osc f|_{A} ≡ ess sup_{ω∈A} f(ω) − ess inf_{ω∈A} f(ω) .

This condition is too strong to characterize norm compactness.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 47B38, 46B20, 28B99.

This paper is part of the Ph.D. work of the author, who is grateful for the guidance of her advisor Professor J. J. Uhl, Jr.

MARIA GIRARDI 2

Bourgain showed that the Bourgain property, a weakening of the above condition, also guarantees relative norm compactness of relatively weakly compact subsets in L_{1}. A subset K of L_{1} has the \underline{Bourgain property} if for each ϵ > 0 and subset B of [0, 1] with positive measure there is a finite collection F of subsets of B, each with positive measure, such that for each f ∈ K there is an A in F with osc f|_{A} < ϵ . However, this condition also is too strong to characterize norm compactness.

We introduce an L_{1}-oscillation that is a weakening of the L_{∞}-oscillation used in the above conditions. The \underline{Bocce oscillation} of an L_{1}-function f on a subset A is given by

Bocce-osc f|_{A} ≡ \frac{∫_{A} |f − ^{\frac{∫_{A} f dμ}{μ(A)}}| dμ}{μ(A)} ,

observing the convention that 0/0 is 0. A subset K of L_{1} satisfies the \underline{Bocce criterion} if for each ϵ > 0 and subset B of [0, 1] with positive measure there is a finite collection F of subsets of B, each with positive measure, such that for each f ∈ K there is an A in F with Bocce-osc f|_{A} < ϵ . The Bocce criterion is a weakening of the Bourgain property. The main result of this paper is that a relatively weakly compact subset of L_{1} is relatively L_{1}-norm compact if and only if it satisfies the Bocce criterion.

Throughout this paper, 𝔛 denotes an arbitrary Banach space, 𝔛^{∗} the dual space of 𝔛, B(𝔛) the closed unit ball of 𝔛, and S(𝔛) the unit sphere of 𝔛. The triple (Ω, Σ, μ) refers to the Lebesgue measure space on [0, 1], Σ^{+} to the sets in Σ with positive measure, L_{1} to L_{1}(Ω, Σ, μ). Recall that a subset of L_{1} is relatively weakly compact if and only if it is bounded and uniformly integrable. All notation and terminology, not otherwise explained, are as in [DU].

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 3

2 : Main Result

Call a subset K of L_{1} a \underline{set of small Bocce oscillation} if for each ϵ > 0 there is a finite positive measurable partition P of Ω such that for each f in K

∑_{A∈P} μ(A) Bocce-osc f|_{A} < ϵ .

Bocce oscillation, the Bocce criterion and norm compactness in L_{1} are related by Theorem 2.1, the main theorem of this paper. Theorem 2.1. For a relatively weakly compact subset K of L_{1}, the following statements are equivalent.

(1) K is relatively norm compact.

(2) K is a set of small Bocce oscillation.

(3) K satisfies the Bocce criterion.

Proof. It is well-known and easy to check that a bounded subset K of L_{1} is relatively norm compact if and only if for each ϵ > 0 there is a finite measurable partition π of Ω such that ∥ f − E_{π}(f) ∥_{L_{1}}< ϵ for each f in K. Here, E_{π}(f) denotes the conditional expectation of f relative to the sigma field generated by π. The equivalence of (1) and (2) follows directly from this observation, the computation below,

∥ f − E_{π}(f) ∥_{L_{1}} = ∫_{Ω} | f − ∑_{B∈π} \frac{∫_{B} f dμ}{μ(B)}χ_{B} | dμ

= ∑_{B∈π} ∫_{B} | f − \frac{∫_{B} f dμ}{μ(B)} | dμ = ∑_{B∈π} μ(B) Bocce-osc f|_{B} ,

and the definition of a set of small Bocce oscillation.

Viewed as a function from Σ into the real numbers, Bocce-osc f|_{(·)} is not increasing (see Example 2.3); however, μ(·) Bocce-osc f|_{(·)} is increasing (see Remark 2.4). With this in mind, we now show that (2) implies (3) in Theorem 2.1.

MARIA GIRARDI 4

Let the subset K of L_{1} be a set of small Bocce oscillation. Fix ϵ > 0 and B in Σ^{+}. Since K is a set of small Bocce oscillation, there is a positive measurable partition π = {A_{1}, A_{2}, …, A_{n}} of Ω such that for each f in K

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

Set F = {A_{i} ∩ B : A_{i} ∈ π and μ(A_{i} ∩ B) > 0}.

Fix f in K. Since the function μ(·) Bocce-osc f|_{(·)} is increasing, if Bocce-osc f|_{A_{i}∩B} ≥ ϵ for each set A_{i} ∩ B in F then we would have that

ϵ · μ(B) = ∑_{i=1}^{n} ϵ · μ(A_{i} ∩ B) ≤ ∑_{i=1}^{n} μ(A_{i} ∩ B) Bocce-osc f|_{A_{i}∩B}

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

This cannot be, so there is a set A_{i} ∩ B in F such that the Bocce-osc f|_{A_{i}∩B} < ϵ. Thus the set K satisfies the Bocce criterion.

We need the following lemma which will will prove after the proof of Theorem 2.1. This lemma is the key step in the proof that (3) implies (1). Lemma 2.2. Let the relatively weakly compact subset K of L_{1} satisfy the Bocce criterion. If ϵ > 0 and {f_{n}} is a sequence in K, then there are disjoint sets B_{1}, …, B_{p} in Σ^{+} and a subsequence {g_{n}} of {f_{n}} satisfying

∫_{Ω} | g_{n} − ∑_{j=1}^{p} \frac{∫_{B_{j}} g_{n} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ 2 ϵ .  (∗∗)

We now proceed with showing that (3) implies (1) in Theorem 2.1. Let the relatively weakly compact subset K of L_{1} satisfy the Bocce criterion. Choose a sequence {f_{n}} in K and a sequence {ϵ_{k}} of positive real numbers decreasing to zero.

It suffices to find a sequence {π_{k}} of finite measurable partitions of Ω and a nested sequence {{f_{n}^{k}}_{n}}_{k} of subsequences of {f_{n}} such that for all positive integers n and k

∥ f_{n}^{k} − E_{k}(f_{n}^{k}) ∥_{L_{1}} ≤ 2 ϵ_{k} ,  (∗)

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 5

where E_{k}(f) denotes the conditional expectation of f relative to the σ-field generated by π_{k}. For then the set {f_{n}^{n}} is relatively L_{1}-norm compact since it can be uniformly approximated in the L_{1}-norm within 2ϵ_{k} by the relatively compact set {E_{k}(f_{n}^{n})}_{n≥k} ∪ {f_{n}^{n}}_{n<k}; hence, there is a L_{1}-norm convergent subsequence of {f_{n}}.

Towards (∗), repeated applications of Lemma 2.2 yield that for each positive integer k:

(1) disjoint subsets B_{1}^{k}, …, B_{n_{k}}^{k} of Ω each with positive measure, and

(2) a subsequence {f_{n}^{k}}_{n} of {f_{n}^{k−1}}_{n} (where we write {f_{n}^{0}} for {f_{n}} ) satisfying for each positive integer n

∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ ≤ ϵ_{k} .

Set

B_{0}^{k} = Ω \ ⋃_{j=1}^{n_{k}} B_{j}^{k}

and let π_{k} be the partition generated by B_{0}^{k}, B_{1}^{k}, …, B_{n_{k}}^{k}. Still observing the convention that 0/0 is 0, two appeals to the above inequality yield for each positive integer n and k

∥ f_{n}^{k} − E_{k}(f_{n}^{k}) ∥_{L_{1}} = ∫_{Ω} | f_{n}^{k} − ∑_{j=0}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ ∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ + ∫_{Ω} | \frac{∫_{B_{0}^{k}} f_{n}^{k} dμ}{μ(B_{0}^{k})}χ_{B_{0}^{k}} | dμ

≤ ϵ_{k} + ∫_{B_{0}^{k}} | f_{n}^{k} | dμ

= ϵ_{k} + ∫_{B_{0}^{k}} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ ϵ_{k} + ∫_{Ω} | f_{n}^{k} − ∑_{j=1}^{n_{k}} \frac{∫_{B_{j}^{k}} f_{n}^{k} dμ}{μ(B_{j}^{k})}χ_{B_{j}^{k}} | dμ

≤ 2 ϵ_{k} .

MARIA GIRARDI 6

Thus the proof of Theorem 2.1 is finished as soon as we verify Lemma 2.2. Proof of Lemma 2.2. Let the relatively weakly compact subset K of L_{1} satisfy the Bocce criterion. Fix ϵ > 0 and a sequence {f_{n}} in K. The proof is an exhaustiontype argument.

Let E_{1} denote the collection of subsets B of Ω such that there are disjoint subsets B_{1}, …, B_{p} of B each with positive measure and a subsequence {f_{n_{k}}} of {f_{n}} satisfying

∫_{B} | f_{n_{k}} − ∑_{j=1}^{p} \frac{∫_{B_{j}} f_{n_{k}} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ μ(B) .

Since K satisfies the Bocce criterion, there is an A in Σ^{+} and a subsequence {f_{n_{k}}} of {f_{n}} satisfying

∫_{A} | f_{n_{k}} − \frac{∫_{A} f_{n_{k}} dμ}{μ(A)} | dμ ≤ ϵ μ(A) .

Thus the collection E_{1} is not empty. If there is a set B in E_{1} with corresponding subsets B_{1}, …, B_{p} and a subsequence {g_{n}} of {f_{n}} satisfying

∫_{Ω} | g_{n} − ∑_{j=1}^{p} \frac{∫_{B_{j}} g_{n} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ ,  (1)

then we are finished since (1) implies (∗∗).

Otherwise, let j_{1} be the smallest positive integer for which there is a C_{1} in E_{1} with \frac{1}{j_{1}} ≤ μ(C_{1}). Accordingly, there is a finite sequence \{C_{j}^{1}\} of disjoint subsets of C_{1} each with positive measure and a subsequence {f_{n}^{1}} of {f_{n}} satisfying

∫_{C_{1}} | f_{n}^{1} − ∑_{j} \frac{∫_{C_{j}^{1}} f_{n}^{1} dμ}{μ(C_{j}^{1})}χ_{C_{j}^{1}} | dμ ≤ ϵ μ(C_{1}) .  (2)

Note that since condition (1) was not satisfied, μ(C_{1}) < μ(Ω) = 1 .

Let E_{2} denote the collection of subsets B of Ω \ C_{1} such that there are disjoint subsets B_{1}, …, B_{p} of B each with positive measure and a subsequence {f_{n_{k}}} of

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 7

{f_{n}^{1}} satisfying

∫_{B} | f_{n_{k}} − ∑_{j=1}^{p} \frac{∫_{B_{j}} f_{n_{k}} dμ}{μ(B_{j})}χ_{B_{j}} | dμ ≤ ϵ μ(B) .

Since K satisfies the Bocce criterion and μ(Ω\C_{1}) > 0, we see that E_{2} is not empty. If there is a set B in E_{2} with corresponding finite sequence {C_{j}^{2}} of subsets of B and a subsequence {g_{n}} of {f_{n}^{1}} satisfying

∫_{Ω\C_{1}} | g_{n} − ∑_{j} \frac{∫_{C_{j}^{2}} g_{n} dμ}{μ(C_{j}^{2})}χ_{C_{j}^{2}} | dμ ≤ ϵ μ(Ω \ C_{1}) ,  (3)

then we are finished. For in this case, (∗∗) holds since inequalities (2) and (3) insure that

∫_{Ω} | g_{n} − ∑_{\begin{matrix}k=1,2 \\ j \end{matrix}} \frac{∫_{C_{j}^{k}} g_{n} dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ ≤ ϵ μ(C_{1}) + ϵ μ(Ω \ C_{1}) = ϵ .

Otherwise, let j_{2} be the smallest positive integer for which there is a C_{2} in E_{2} with \frac{1}{j_{2}} ≤ μ(C_{2}). Accordingly, there is a finite sequence \{C_{j}^{2}\} of disjoint subsets of C_{2} each with positive measure and a subsequence {f_{n}^{2}} of {f_{n}^{1}} satisfying

∫_{C_{2}} | f_{n}^{2} − ∑_{j} \frac{∫_{C_{j}^{2}} f_{n}^{2} dμ}{μ(C_{j}^{2})}χ_{C_{j}^{2}} | dμ ≤ ϵ μ(C_{2}) .

Note that since condition (3) was not satisfied, μ(C_{2}) < μ(Ω \ C_{1}) .

Continue in this way. If the process stops in a finite number of steps then we are finished. If the process does not stop, then diagonalize the resultant sequence { {f_{n}^{k}}_{n=1}^{∞} }_{k=1}^{∞} of sequences to obtain the sequence {f_{n}^{n}}_{n=1}^{∞} and set C_{∞} = ⋃_{k=1}^{∞} C_{k} .

Note that μ(Ω \ C_{∞}) = 0. For if μ(Ω \ C_{∞}) > 0 then, since K satisfies the Bocce criterion, there is a subset B of Ω \ C_{∞} with positive measure and a subsequence {h_{n}} of {f_{n}^{n}} satisfying

∫_{B} | h_{n} − \frac{∫_{B} h_{n} dμ}{μ(B)} | dμ < ϵ μ(B) .

Since for each positive integer m

∑_{k=1}^{m} \frac{1}{j_{k}} ≤ μ( ⋃_{k=1}^{m} C_{k}) ≤ μ(Ω)

and j_{m} > 1, we can choose an integer m > 1 such that

\frac{1}{j_{m} − 1} ≤ μ(B) .

But this implies that B is in E_{m} since

B ⊂ Ω \ C_{∞} ⊂ Ω \ ⋃_{k=1}^{m−1} C_{k}

and {h_{n}}_{n=m−1}^{∞} is a subsequence of {f_{n}^{n}}_{n=m−1}^{∞}, which in turn is a subsequence of {f_{n}^{m−1}}_{n=1}^{∞}. This contradicts the choice of j_{m}. Thus μ(Ω \ C_{∞}) = 0.

Since K is relatively weakly compact, it is uniformly integrable. Thus, there exists δ > 0 such that if μ(A) < δ then ∫_{A} | f | dμ < ϵ for each f ∈ K. Pick an integer m so that

μ(Ω \ ⋃_{k=1}^{m} C_{k}) < δ .

Since {f_{n}^{n}}_{n=m}^{∞} is a subsequence of {f_{n}^{m}}_{n=1}^{∞}, for each f in {f_{n}^{n}}_{n=m}^{∞} we have

∫_{Ω} | f− ∑_{\begin{matrix}1≤k≤m \\ j \end{matrix}} \frac{∫_{C_{j}^{k}} f dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ

= ∑_{k=1}^{m} ∫_{C_{k}} | f − ∑_{j} \frac{∫_{C_{j}^{k}} f dμ}{μ(C_{j}^{k})}χ_{C_{j}^{k}} | dμ + ∫_{Ω\⋃_{k=1}^{m} C_{k}} |f| dμ

≤ ∑_{k=1}^{m} ϵ μ(C_{k}) + ϵ ≤ 2 ϵ .

So the disjoint subsets {C_{j}^{k} : j ≥ 1 and k = 1, …, m} along with the subsequence {f_{n}^{n}}_{n=m}^{∞} of {f_{n}} satisfy the conditions of the lemma.

This completes the proof of Theorem 2.1.

We close this section with a few observations about Bocce oscillation.

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 9

Example 2.3. Let A = [0, 1/4] and B = [0, 1]. Define the function f from [0,1] into the real numbers by f(t) = χ_{C}(t) where C = [1/8, 1]. It is straightforward to verify that Bocce-osc f|_{A} = 1/2 but Bocce-osc f|_{B} = 7/32. Remark 2.4. In the proof of Theorem 2.1, we used the fact that the function

μ(·) Bocce-osc f|_{(·)} ≡ ∫_{(·)} | f − \frac{∫_{(·)} f dμ}{μ(·)} | dμ

is an increasing function from Σ into the reals numbers. To see this, let A be a subset of B with A and B in Σ^{+}. Let

m_{A} = \frac{∫_{A} f dμ}{μ(A)} and m_{B} = \frac{∫_{B} f dμ}{μ(B)} .

Note that

∫_{A} | f − m_{A} | dμ ≤ ∫_{A} | f − m_{B} | dμ + ∫_{A} | m_{B} − m_{A} | dμ
= ∫_{B} | f − m_{B} | dμ − ∫_{B\A} | f − m_{B} | dμ + ∫_{A} | m_{B} − m_{A} | dμ ,

and

∫_{A} | m_{B} − m_{A} | dμ = μ(A) | m_{B} − m_{A} |

= | \frac{μ(A)}{μ(B)} ∫_{B} f dμ − ∫_{A} f dμ |

= | \frac{μ(A)}{μ(B)} ∫_{B} f dμ − (∫_{B} f dμ − ∫_{B\A} f dμ) |

= | ∫_{B\A} f dμ − \frac{μ(B) − μ(A)}{μ(B)} ∫_{B} f dμ |

= | ∫_{B\A}(f − m_{B}) dμ | .

Thus, as needed,

∫_{A} | f − m_{A} | dμ ≤ ∫_{B} | f − m_{B} | dμ .

MARIA GIRARDI 10

Remark 2.5. The Bocce oscillation has been implicitly studied. By definition, a function f in L_{1} is of BMO (bounded mean oscillation) provided sup_{I} Bocce-osc f|_{I} < ∞ where the sup is over all intervals contained in [0, 1]. 3 : Applications

Fix a bounded linear operator T from L_{1} into 𝔛 and consider the uniformly bounded subset T^{∗}(B(𝔛^{∗})) = { T^{∗}(x^{∗}) : ∥ x^{∗} ∥ ≤ 1} of L_{1}. The oscillation behavior of elements in T^{∗}(B(𝔛^{∗})) provides information about T.

Ghoussoub, Godefroy, Maurey, and Schachermayer [GGMS] showed that the operator T is strongly regular if and only if the subset T^{∗}(B(𝔛^{∗})) of L_{1} has the Bourgain property. Recall that an operator is called Dunford-Pettis if it maps weakly convergent sequences onto norm convergent sequences. With Theorem 2.1, we obtain an analogous characterization of Dunford-Pettis operators. From the oscillation characterizations, it is easy to see that a strongly regular operator is Dunford-Pettis. Corollary 3.1. A bounded linear operator T from L_{1} into a Banach space 𝔛 is Dunford-Pettis if and only if the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion. Corollary 3.1 follows directly from Theorem 2.1 and the fact below. Fact 3.2. The following statements are equivalent.

(1) T is a Dunford-Pettis operator.

(2) T maps weak compact sets to norm compact sets.

(3) T (B(L_{∞})) is a relatively norm compact subset of 𝔛.

(4) The adjoint of the restriction of T to L_{∞} from 𝔛^{∗} into L_{∞}^{∗} is a compact operator.

(5) As a subset of L_{1}, T^{∗}(B(𝔛^{∗})) is relatively norm compact.

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 11

The equivalence of (2) and (3) follows from the fact that the subsets of L_{1} that are relatively weakly compact are precisely those subsets that are bounded and uniformly integrable, which in turn, are precisely those subsets that can be uniformly approximated in L_{1}-norm by uniformly bounded subsets. The other implications are easy to verify [cf. DU].

Ghoussoub, Godefroy, Maurey, and Schachermayer also obtained an internal geometric description of the class of Banach spaces with the property that all bounded linear operators from L_{1} into 𝔛 are stongly regular. Using Corollary 3.1, we obtain internal geometric descriptions of the class of Banach spaces with the property that all bounded linear operators from L_{1} into 𝔛 are Dunford-Pettis. To provide a flavor of the techniques involved in these descriptions, we present an outline of one the arguments.

Theorem 3.3. If each bounded subset of a Banach space 𝔛 is weak-norm-one dentable, then each bounded linear operator from L_{1} into 𝔛 is Dunford-Pettis.

Recall that subset D is weak-norm-one dentable if for each ϵ > 0 there is a finite subset F of D such that for each x^{∗} in S(𝔛^{∗}) there is x in F satisfying

x ∉ \overline{co} (D \ V_{ϵ,x^{∗}}(x)) ≡ \overline{co} \{z ∈ D : | x^{∗}(z − x) | ≥ ϵ\} .

Sketch of Proof. Let all bounded subsets of 𝔛 be weak-norm-one dentable. Fix a bounded linear operator T from L_{1} into 𝔛. By Corollary 3.1, it is enough to show that the subset T^{∗}(B(𝔛^{∗})) of L_{1} satisfies the Bocce criterion.

To this end, fix ϵ > 0 and B in Σ^{+}. Let F denote the vector measure from Σ into 𝔛 given by F (E) = T (χ_{E}). Since the subset \{\frac{F (E)}{μ(E)} : E ⊂ B and E ∈ Σ^{+}\} of 𝔛 is bounded, it is weak-norm-one dentable. Accordingly, there is a finite collection F of subsets of B each in Σ^{+} such that for each x^{∗} in the unit ball of 𝔛^{∗} there is a set A in F such that if

\frac{F (A)}{μ(A)} = \frac{1}{2} \frac{F (E_{1})}{μ(E_{1})} + \frac{1}{2} \frac{F (E_{2})}{μ(E_{2})}

for some subsets E_{i} of B with E_{i} ∈ Σ^{+}, then

\frac{1}{2} |\frac{x^{∗}F (E_{1})}{μ(E_{1})} − \frac{x^{∗}F (A)}{μ(A)}| + \frac{1}{2} |\frac{x^{∗}F (E_{2})}{μ(E_{2})} − \frac{x^{∗}F (A)}{μ(A)}| < ϵ .

Fix x^{∗} ∈ B(𝔛^{∗}) and find the associated A in F. By definition, the set T^{∗}(B(𝔛^{∗})) will satisfy the Bocce criterion provided that Bocce-osc (T^{∗}x^{∗})|_{A} ≤ ϵ.

For a finite positive measurable partition π of A, denote

f_{π} = ∑_{E∈π} \frac{F (E)}{μ(E)}χ_{E} .

Set

E_{π}^{+} = ⋃\{ E ∈ π : \frac{x^{∗}F (E)}{μ(E)} ≥ \frac{x^{∗}F (A)}{μ(A)} \} ,

and

E_{π}^{−} = ⋃\{ E ∈ π : \frac{x^{∗}F (E)}{μ(E)} < \frac{x^{∗}F (A)}{μ(A)} \} .

We may assume that the L_{1}-function T^{∗}x^{∗} is not constant a.e. on A. Furthermore, we may find an increasing sequence {π_{n}} of positive measurable partitions of A satisfying

∨ σ(π_{n}) = Σ ∩ A  and  μ(E_{π_{n}}^{+}) = \frac{μ(A)}{2} = μ(E_{π_{n}}^{−}) .

It is easy to verify that since F (A) / μ(A) has the proper form,

∫_{A} |x^{∗}f_{π_{n}} − \frac{x^{∗}F (A)}{μ(A)}| dμ

= μ(A)[ \frac{1}{2} |\frac{x^{∗}F (E_{π_{n}}^{+})}{μ(E_{π_{n}}^{+})} − \frac{x^{∗}F (A)}{μ(A)}| + \frac{1}{2} |\frac{x^{∗}F (E_{π_{n}}^{−})}{μ(E_{π_{n}}^{−})} − \frac{x^{∗}F (A)}{μ(A)}| ]

< μ(A) ϵ .

COMPACTNESS IN L_{1}, D-P OPERATORS, GEOMETRY OF BANACH SPACES 13

Since (x^{∗}f_{π_{n}})|_{A} converges to (T^{∗}x^{∗})|_{A} in L_{1}-norm ,

Bocce-osc (T^{∗}x^{∗})|_{A} ≡ \frac{∫_{A} |(T^{∗}x^{∗}) − ^{\frac{∫_{A}(T^{∗}x^{∗}) dμ}{μ(A)}} | dμ}{μ(A)} ≤ ϵ.

Thus T^{∗}(B(𝔛^{∗})) satisfies the Bocce criterion, as needed.

Using martingale techniques, one can show that the converse of Theorem 3.3 is

true. For this argument and more results along these line, we refer the reader to [G].

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Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801