For a real number t Î [0,1], we call a set A Ì R2 a t-shading of the plane (or a set of shade t) if m(AÇE) = t m(E) whenever E Ì R2 is Lebesgue measurable and m is a Banach measure on R2, i.e., m is an isometry-invariant extension of the Lebesgue measure defined for all subsets of the plane. We then write sh2(A) = t. A t-shading A of R is defined in an analagous fashion, and we write sh1(A) = t. We may also consider one-dimensional Banach measures defined on an arbitrary line L Ì R2. If AÇL is a one-dimensional t-shading, we write shL(A) = t. (See [5] for proofs of the existence of t-shadings in R and R2.)
As an intuitive aid, we will consider a t-shading of R2 (resp., R) to be a set for which the probability of hitting the set by a point-dart thrown at R2 (resp., R) is t.
The following classic example may help motivate the rest of the discussion. In 1920, Sierpi\'nski showed that CH implies that
In terms of throwing darts, we conclude that darts thrown in horizontal directions will almost surely miss S, while darts thrown in vertical directions almost surely strike S. Expressed in other terms, (S) illustrates that there is no strong Fubini theorem under CH. (Other set-theoretic axioms, independent of ZFC and weaker than CH, also yield (S) . See, e.g., [1], [2], [6], [7].)
Lest one be tempted to reject CH in view of the above, we note that (S) is a theorem of ZFC+CH, so AC may be as much to blame3. In fact, one may take any well-ordering Ð of R and let S = {(x,y) Î R2:xÐy}. Thus a horizontal section, for any fixed y, has cardinality < c while the complements of vertical sections also have this property. If CH holds, these sets are countable, and we certainly then have that shL(S) = 0 if L is a horizontal line, while shL(S) = 1 if L is vertical. On the other hand it is a fact in ZFC that if X Ì R and |X| < c, then sh1(X) = 0 (see [5]). Thus ZFC is sufficient for such a paradox (in the sense of throwing darts).
The following shows that densities of linear sections of a plane set are even more independent than is suggested by (S) .
Proposition 1 Let L denote the set of lines in R2 and let f:L® [0,1] be any function whatsoever. Then there exists A Ì R2 with the property that for any line L Ì R2, the set LÇA is an f(L)-shading of L.
Sketch of proof. Injectively well-order L as (Lb)b < c. For each b < c, let Bb be any f(Lb)-shading of Lb. If for b < c the set Ag has already been constructed for each g < b, then let Ab = Bb \Èg < bLg. The set A = Èb < cAb has the desired properties.
In particular, f(L) may have a constant value t Î (0,1).
Problem 2 Suppose that A is a subset of the plane whose linear shades are all equal to t. What, if anything, can be said about the planar shade of A? In particular, is it possible for sh2(A) = s ¹ t?
Being unable to answer the above question, at least we can do this much:
Proposition 3
For each t Î [0,1] there exists T Ì R2 for which
|
The set T = f-1(D) does the trick. Property (1) follows from the fact that for all nonempty open sets U in C, the set f-1(U)ÇL is dense in every line L Ì C. The additive property of f can be used to prove property (2).
My thanks go to the organizers of the Andy Symposium for an
outstanding conference, and to Andy himself for his kind help and
great sense of humor.4
1 This presentation has been modified from its original form. It has been edited, sanitized, and for the most part stripped of foolish and irreverent comments, in order to fit this journal.
2 The author thanks Alexander Kharazishvili(Tbilisi) and Chris Ciesielski (Morgantown) for their helpful correspondence.
3 In the live version, the speaker half-jokingly chided his friends in the real analysis community for not rejecting any set-theoretic axioms which allow (S) and accused them of living in ``hip denial'' of such paradoxes.
4 To ``wit'': Referring to the paper [A], the speaker wondered aloud whether its authors had actually succeeded in improving Lebesgue measure. From the audience, Andy quickly affirmed, ``Oh yes, it's much better now!''