Mathematical logic seminar - May 3, 2011

Time: 12:00 - 13:20

Room: Wean Hall 7201

Speaker:     Jason Rute    
Department of Mathematical Sciences
Carnegie Mellon University

Title: Randomness and the Lebesgue Differentiation Theorem.


The Lebesgue Differentiation Theorem on $[0,1]^n$ states that, given any function $f$ integrable with respect to the Lebesgue measure $m$, then for almost any $x \in [0,1]^n$, the average $\int_{B_r (x)}f\, dx/m(B_r (x))$ converges to $f(x)$. We show that it is possible to characterize the $x$ for which the above theorem holds as follows: If $f$ is an $L^{1}$-computable function $f$ and $x$ is a Schnorr random real, then the above fraction converges for all such sequences of intervals. Further, if $x$ is not Schnorr random, then there is some $L^{1}$-computable $f$ such that the fraction diverges. This answers a question of Pathak, who asked what type of randomness the Lebesgue differentiation theorem characterizes. It also contributes to a program of Brattka, Miller, and Nies to characterize randomness in terms of differentiability.