Differentiability and Measures in Banach Spaces
David Preiss
Department of Mathematics, University College London, London WC1E 6BT, UK
The purpose of this contribution is to give information about new results concerning natural questions about differentiability and measures in real Banach
spaces (of infinite but also of finite dimension) and, possibly more importantly,
to point out some of the many open problems we are still faced with in this area
of research.
1. Differentiability
We recall two well known notions.
1. A real valued function / defined on an open subset G of a Banach space
E is said to be Fréchet differentiate at a point x G G if there is f(x) e E* such
that
lim
«-0
\f(x + u)-f(x)-(f(x),u)\
= 0
||u||
f(x) is called the Fréchet derivative of/ at x.
2. A real valued function / defined on an open subset G of a Banach space E is
said to be Lipschitz on G if there is a constant C such that \f(x)— f(y)\ < C\\x—y\\
whenever x,y G G. The smallest such constant C is denoted by Lip(/).
From the work of Lebesgue (in the one dimensional case) and of Rademacher
(in the finite dimensional case) we know that Lipschitz functions on finite dimensional spaces are (Fréchet) differentiate almost everywhere with respect to
the Lebesgue measure. Infinite dimensional results of similar nature are known
for Gateaux differentiability. (See [1,3,5,6]). These extension are obtained by a
linear approximation of the infinite dimensional situation by finite dimensional
spaces. However, the question of Fréchet differentiability seems to need a different approach. This might be also seen from many examples of nowhere Fréchet
differentiable Lipschitz mappings of a separable Hilbert space into itself, since for
such mappings the Gateaux differentiability results mentioned above still hold.
Thus our first result answers a natural question.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
924
David Preiss
Theorem 1. Every Lipschitz function defined.on a separable Hilbert] space is Fréchet
differentiate at least at one point.
Hilbert spaces are, of course, not the most general spaces in which one would
hope for such a result. Indeed, from the extensive investigations of differentiability
questions for continuous convex functions (e.g., [13,14]) we know that the result
may hold in all Asplund spaces. (A Banach space is said to be an Asplund space
if the dual of every its separable subspace is separable.) This generalization of
Theorem 1 is given in the following statement.
Theorem 2. Every locally Lipschitz function defined on an open subset of an Asplund
space is Fréchet dijferentiable on a dense subset of its domain.
The method we use need not be confined to Fréchet differentiability. It
applies also to so called & derivatives, in the definition of which we require
the uniform convergence on the members of a given family ^ of bounded
subsets of the Banach space satisfying some mild additional assumptions. (The
details can be found in [10].) This gives the most general form of the above
differentiability results. (However, a recent Haydon's example of an Asplund
space without equivalent smooth norms shows that the deduction of Theorem 2
from Theorem 3 is not straightforward.)
Theorem 3. Let E be a Banach space admitting an equivalent norm which is M
dijferentiable away from the origin. Then every locally Lipschitz function defined
on an open subset G of E is & differentiable on a dense subset of G.
These statements, as given, are not satisfactory from the point of view of
possible applications. For example, suppose that a Lipschitz. function / on a
separable Hilbert space has derivative zero at every point at which it is Fréchet
differentiable. We would like to be able to deduce that / is constant. This can be
done, since in all the above results the mean value theorem holds. For example, in
case of Theorem 3 we prove that the increment of the function over any segment
[u, v] c: G is majorized by the supremum of the derivatives in the direction v — u
at points at which the function is & differentiable.
The proof of the above results requires new information about Lipschitz
functions in finite dimensional spaces. Thus, as a byproduct, we get the following
curious statement.
Theorem 4. There is a piarle set N of Lebesgue measure zero such that every Lipschitz function defined on the plane is differentiable at some point of N.
To describe a set having such a property is quite easy: Any Gs plane set of
Lebesgue measure zero containing all lines passing through two different points
with rational coordinates will do. This particular example also suggests the
reasons why our proof of Fréchet differentiability results is not straightforward.
It combines in some way two notions of smallness of a set: First category (hence
the Gs part) and measure zero (hence the lines). It seems to be intuitively clear
that a similar mixture is impossible on the line. That this is true has been shown
in [2] and [15] : Theorem 4 is false on the line..
Differentiability and Measures in Banach Spaces
925
1.1 Construction of a Point of Fréchet Differentiability
The details of the proof can be found in [10]. Here we just point out the main
observations. Because of that we restrict our attention to the proof of Theorem 1
only.
We introduce the directional derivatives of/ by
f'(x,e) = limf{x
r->0
+
re) f{x
- \
r
and we denote by M the set of all pairs (x,e) e E x E* such that \\e\\ = 1 and
ff(x,e) exists.
The first basic observation is that if (x,e) e M and f(x,e) = Lip(/) then
/ is Fréchet differentiable at x. Even though such a pair need not exist, this
suggests that we might attempt to use a maximizing procedure. Thus our plan is
to construct inductively a sequence (xk,eu) G M so that:
1.
2.
3.
4.
The sequence Xk converges to some x.
The sequence e/c converges to some e.
The directional derivative f(x,e) exists.
For the pair (x, e) some variant of the above observation can be used.
To achieve 1, we simply choose x^+i close to x*. This is based on a local
form of our observation, namely, that the equality of f'(x,e) to the limit of the
Lipschitz constants on balls around x suffices for Fréchet differentiability of / at
x.
Unfortunately, to get 2 is not so simple. Since requirement 4 forces us to
take /'(xfcjß/c) as large as possible, we cannot at the same time prescribe how
close should ek be to e^-\. There is also a different objection we should take
into account: If our method worked, we would construct not only a point of
differentiability, but also a point at which gradient vector exists. This causes no
problem in Hilbert spaces, but is impossible in non-reflexive Asplund spaces.
(Every liner functional not attaining its maximum on the unit ball gives an
example.) Thus an idea suggests itself: We should change the norm (and the
change should depend on / ) , at least in the general case. Recalling that we are
constructing a sequence e^ of unit vectors, and observing that a small change
of the norm can drastically change the set of pairs considered for the choice of
(x/c+i,e/c+i), we find that 2 can be achieved by constructing, together with the
sequence (x/c, ej(), a sequence of norms p/c, where pu+i is the (e.g., h) sum of pk
and of a (small) multiple of the distance to the one dimensional subspace of H
generated by ejc. Then the conditions p*+i(e/H-i) = 1 (= p(eu)) and f(xk+uek+i) >
/'(^/CJ^/C) already imply that ej<+i is close to ejc.
The requirement 3 seems to be the most difficult. To get it, we observe that the
problem is essentially one dimensional and requires some method of interchange
of limit and derivative. Since we cannot hope to be able to use anything like
the uniform convergence of the derivatives, the only possibility seems to be to
choose the points at which the increment of the function is approximated by the
derivative globally. The following one dimensional lemma says that this can be
done.
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David Preiss
Lemma 5. Suppose that a < Ç < b, 0 < a < 1/4, and L > 0 are real numbers, h is
a Lipschitz function defined on [a,b], Lip(/i) < L, h(a) = h(b) — 0, and h(Ç) ^ 0.
Then there is a measurable set A c (a, b) such that
1.
2.
3.
The Lebesgue measure of the set A is at least a\h(t;)\/L,
h'(x) > (T\h(Ç)\/(b -v- a) for every % e A, and
\h(t) — h(n)\ < 4(1 + 2a)^h'(x)L\t — T| for every % e A and every t e [a,b].
The most important third statement of the lemma says that the approximation
of the increment of the function by its derivative at the point % is "globally good"
in the whole interval [a,b]. The second statement just says that the derivative at
T increased as much as we could hope for. From the first statement we just use
that T can be chosen sufficiently far from the end points. This is needed in order
to get a bilateral approximation.
Because in the first statement of Lemma 5 we do not have to speak about
measure, the Lemma can be formulated without the notion of the Lebesgue
measure. We can then try to prove it without any use of measure theory. This
sounds difficult, since we also claim that h is differentiable at x. But we can also
replace the derivatives by lower derivatives and get a version of the lemma that
really can be proved without any use of measure theory. Surprisingly enough,
this statement then easily implies that Lipschitz functions on the real line have at
least one point of differentiability. Though I did not follow this way, since to use
the Lebesgue measure and maximal operator technique turned out to be much
easier, these remarks suggest that the proof of differentiability discussed here is
different from the usual measure theoretic proofs.
Having done this, we can already imagine how to construct the sequence
(x/c,ßfc) so that 3 holds: We will choose (xk+uCk+i) so that the approximation of
the the increment of the function by its directional derivative at the point xj^+i
in the direction e^i is "globally good" on the whole Une through Xfc+i in the
direction ek+\.
However, the previous choice implies that our construction will lead to a pair
(x,e) for which the equality f'(x,e) = Lip(/) is quite far from being true. Hence
to achieve 4 we need to improve upon our main observation. We first reformulate
this observation as:
A Lipschitz function / on a Hilbert space is Fréchet differentiable at x if
there is a unit vector e such that f(x,e) exists and
lim sup{/'(x, e); (x, e) e M and ||x — x|| < 3} < f(x, e).
A simple proof of this statement together with Lemma 5 gives the following
differentiability criterion, which we formulate in the most general situation.
Theorem6. Suppose that E is a Banach space, x y G E, eo e E, \\eo\\ = 1, and that f
is a Lipschitz function defined on E such that f'(xo, eo) exists. Let M denote the set
of all pairs (x,e) E Ex{e G E;\\e\\ = 1} such that f(x, e) exists, f(x, e) > f(xo, eo),
and
Differentiability and Measures in Banach Spaces
\(f(x + teo) -f(x))
- (/(xo + too) -/(*>))! <
927
6\tW(ff(x,e)-f'(xo,e0))Liv(f)
for every t G R.
Then, if the norm is & differentiable at eo, and if
\imsSlQsup{f,(x,e);(x,e)
G M and | | x - x 0 | | < ö} </'(xo,ßo),
/ is $ differentiable at xo.
Now, the way of constructing the sequences (xk, e^) and Pk is more or less
clear. We always pick up the next pair from the set M described in the previous
Theorem. The additional requirement is only that /'(x/c+i^+i) is very close to
the supremum of the directional derivatives f'(x,e) for (x,e) G M. Then we define
the norm pk+i and continue our construction. Though we still have to be quite
careful and make some technical estimates, since, for example, the set M from
the previous Theorem depends upon the choice of the norm, this construction
leads to a sequence satisfying all our requirements.
1.2 Problems
From the previous discussion it is clear that the theory of differentiability still
abounds with open problems. I would just like to point out the following two.
Problem 7. Does every pair of Lipschitz functions on a separable Hilbert space
have a common point of differentiability?
Problem 8. For which finite Borei measures in separable Banach spaces is it true
that every Lipschitz function is differentiable almost everywhere?
The second problem is purely finite dimensional since such measures do not
exist in infinite dimensional spaces. (See [12].) The answer is not known in the
plane (or in any higher dimensional space). In the one dimensional case the
required measures are precisely those that are absolutely continuous with respect
to the Lebesgue measure. In spite of Theorem 4 I do not know any example that
would show that this is not true in all finite dimensional spaces.
2. Measures
The question whether measures on separable Banach spaces are determined
by their values on balls has been around since R. O. Davies [4] published his
beautiful example of two different probability measures on a compact metric
space that agree on all balls. Together with J. Tiser [11] we recently answered it
by proving:
Theorem 9. Whenever two finite Borei measures in a separable Banach space agree
on all balls, then they agree.
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David Preiss
To prove this statement, we first use the Fourier transform to reduce the
problem to showing that the measures agree on all halfspaces. Then, by blowing
up balls, we come to the situation when the halfspace contains a nonempty
open cone C on every translate of which the measures agree. An approximation
argument (or a differentiability result from [7]) reduces the problem further to
the case when C n—C is a subspace of finite codimension. Thus we can pass
to the factor space and we have to solve the corresponding problem in finite
dimensional spaces : Do we know the measure of a halfspace provided we know
the measure of each translate of a nonempty open cone contained in it? Since
this turned out to be true, our approach has been successful.
Instead of giving further details, it might be more interesting to point out
some examples. The motivation for them comes from the Besicovitch-Morse
differentiability theorem, which is a much stronger statement than that measures
in finite dimensional normed spaces are determined by their values on balls :
For every (locally) finite Borei measure p in a finite dimensional Banach space
and for every p integrable function / the limit
lim
* xx /
f(u)dp(u)
r^p(B(x,r))JB{Xir)JK)
^ '
(1)
K)
exists and equals f(x) for p alrnost every x.
As a corollary of this statement one can prove that, if p and v are two finite
Borei measures in a finite dimensional Banach space satisfying p(B) > v(B) for
every ball B then p>v.
Example 1 ([9]). There is a Gaussian measure y\ in fe and a yi integrable function
/ such that the limit in (1) is infinite uniformly for x G h, i.e.,
lim inf ——-—— /
f(u) dyAu)
= oo.
m
' r^oxzi2y1(B(x,r))JB{Xir)JK)
'
Example 2 ([8]). There is a Gaussian measure 72 in h and a bounded y2 measurable
function / such that, for p almost every x, the limit in (1) does not exist.
Example 3 (J. Tiser). There is a non-degenerated Gaussian measure y3 in fe such
that (1) holds for every / G Lp(y3), p > 1.
Example 4. In a separable Hilbert space the statement "p(B) > v(B) for balls with
radius less than one implies p > v" holds if and only if the dimension of the
space is finite.
Example 5. In a separable Hilbert space the statement "p(B) > v(B) for balls with
radius greater than one implies p>v" holds if and only if the dimension of the
space is infinite.
Example 6. In the Zoo sum of a separable Hilbert space with the line there are
measures p and v such that p ^ v but p(B) > v(B) for all balls.
Differentiability and Measures in Banach Spaces
929
However, in spite of the above result and examples, the investigation of the
behaviour of measures on balls cannot be considered as finished. For example,
the following question is still far from being answered.
Problem 10. Are finite Borei measures in separable Banach spaces determined by
their values on balls with radii less than one?
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