Neugebauer’s theorem

December 30, 2014

The problem of characterizing when a given function is a derivative has been studied at least since 1911 when Young published a paper recognizing its relevance:

W. H. Young. A note on the property of being a differential coefficient, Proc. Lond. Math. Soc., 9 (2), (1911), 360–368.

The paper opens with the following comment (Young refers to derivatives as “differential coefficients”):

Recent research has provided us with a set of necessary and sufficient conditions that a function may be an indefinite integral, in the generalised sense, of another function, and the way has thus been opened to important developments. The corresponding, much more difficult, problem of determining necessary and sufficient conditions that a function may be a differential coefficient, has barely been mooted; indeed, though we know a number of necessary conditions, no set even of sufficient conditions has to my knowledge ever been formulated, except that involved in the obvious statement that a continuous function is a differential coefficient. 

It is more or less understood nowadays that no completely satisfactory characterization is possible. We know that derivatives are Darboux continuous (that is, they satisfy the intermediate value property), and are Baire one functions (that is, they are the pointwise limit of a sequence of continuous functions). But this is not enough: For instance, any function f such that f(x)=\sin(1/x) for x\ne 0, and f(0)\in[-1,1] is Darboux continuous and Baire one, but only the function with f(0)=0 is a derivative.

Andrew Bruckner has written excellent surveys on derivatives and the characterization problem. See for instance:

Andrew M. Bruckner and John L. Leonard. Derivatives. Amer. Math. Monthly, 73 (4, part II), (1966), 24–56. MR0197632 (33 #5797).

Andrew M. Bruckner. Differentiation of real functions. Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6. MR1274044 (94m:26001).

Andrew M. Bruckner. The problem of characterizing derivatives revisited. Real Anal. Exchange, 21 (1),  (1995/96), 112–133. MR1377522 (97g:26004).

Here I want to discuss briefly a characterization obtained by Neugebauer, see

Christoph Johannes Neugebauer. Darboux functions of Baire class one and derivatives. Proc. Amer. Math. Soc., 13 (6), (1962), 838–843. MR0143850 (26 #1400).

For concreteness, I will restrict discussion to functions f:[0,1]\to\mathbb R, although this makes no real difference. Whenever an interval is mentioned, it is understood to be nondegenerate. For any closed subinterval I\subseteq [0,1], we write I^\circ for its interior, and \mathrm{lh}(I) for its length. Given a point x\in[0,1], we write I\to x to indicate that \mathrm{lh}(I)\to 0 and x\in I.

Theorem (Neugebauer). A function f:[0,1]\to\mathbb R is a derivative iff to each closed subinterval I of {}[0,1] we can associate a point x_I\in I^\circ in such a way that the following hold: 

  1. For all x\in[0,1], if I\to x, then f(x_I)\to f(x), and 

  2. For all  closed subintervals I,I_1,I_2 of {}[0,1], if I=I_1\cup I_2 and {I_1}^\circ\cap {I_2}^\circ=\emptyset, then \mathrm{lh}(I)f(x_I)=\mathrm{lh}(I_1)f(x_{I_1})+\mathrm{lh}(I_2)f(x_{I_2}).

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On strong measure zero sets

December 6, 2013

I meant to write a longer blog entry on strong measure zero sets (on the real line \mathbb R), but it is getting too long, so it may take me more than I expected. For now, let me record here an argument showing the following:

Theorem. If X is a strong measure zero set and F is a closed measure zero set, then X+F has measure zero.

The argument is similar to the one in

Janusz Pawlikowski. A characterization of strong measure zero sets, Israel J. Math., 93 (1), (1996), 171-183. MR1380640 (97f:28003),

where the result is shown for strong measure zero subsets of \{0,1\}^{\mathbb N}. This is actually the easy direction of Pawlikowski’s result, showing that this condition actually characterizes strong measure zero sets, that is, if X+F is measure zero for all closed measure zero sets F, then X is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in measure theory, I am only including here the easy direction.) Pawlikowski’s argument actually generalizes an earlier key result of Galvin, Mycielski, and Solovay, who proved that a set X has strong measure zero iff it can be made disjoint from any given meager set by translation, that is, iff for any G meager there is a real r with X+r disjoint from G.

I proceed with the (short) proof after the fold.

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