Grahl, J.; (2012) Exceptional Lebesgue densities and random Riemann sums. Doctoral thesis , UCL (University College London).

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Abstract

We will examine two topics in this thesis. Firstly we give a result which improved a bound for a question asking which values the Lebesgue density of a measurable set in the real line must have (joint work with Toby O'Neil and Marianna Csörnyei). We also show how this result relates to the results obtained by others. Secondly, we give several results which indicate when a Lebesgue measurable function has a random Riemann integral which converges, in either the weak and strong sense. A Lebesgue measurable set A, subset of R, has density either 0 or 1 at almost every point. Here the density at some point x refers to the proportion of a small ball around x which belongs to A, in the limit as the size of the ball tends to 0. Suppose that A is not either a nullset, which has density 0 at every single point, or the complement of a nullset, which similarly has density 1 everywhere. Then there are certain restrictions on the range of possible values at those exceptional points where the density is neither 0 nor 1. In particular, it is now known that if δ < 0.268486 ..., where the exact value is the positive root of 8δ^3+8 δ^2+ δ1 = 0, then there must exist a point at which the density of A is between δ and 1 - δ, and that this does not remain true for any larger value of δ. This was proved in a recent paper by Ondrej Kurka. Previous to his work our result given in this thesis was the best known counterexample. We give the background to this, construct the counterexample, and discuss Kurka's proof of the exact bound. The random Riemann integral is defined as follows. Given a Lebesgue measurable function f : [0, 1] \rightarrow R and a partition of [0, 1] into disjoint intervals, we can choose a point belonging to each interval, independently and uniformly with respect to Lebesgue measure. We then use these random points to form a Riemann sum, which is itself a random variable. We are interested in knowing whether or not this random Riemann sum converges in probability to some real number. Convergence in probability to r means that the probability that Riemann sum differs from r by more than \varepsilon, is less than \varepsilon, provided that the maximum length of an interval in the partition is sufficiently small. We have previously shown that this type of convergence does take place provided that f is Lebesgue integrable. In other words, the random Riemann integral, defined as the limit in probability of the random Riemann sums, has at least the power of the Lebesgue integral. Here we prove that the random Riemann integral of f does not converge unless |f|^1-e is integrable for e > 0 arbitrarily small. We also give another, more technical, necessary condition which applies to functions which are not Lebesgue integrable but are improper Riemann integrable. We have also done some work on the question of almost sure convergence. This works slightly differently. We must choose, in advance, a sequence of partitions (Pn)n∞=1, with the size of the intervals of Pn tending to zero. We form a probability space on which we can take random Riemann sums independently on each partition of the sequence. Almost sure convergence means that the sequence of random Riemann sums converges to some (unique) limit with probability 1 in this space. There are two complementary results; firstly that almost sure convergence holds if the function is in Lp and the sequence of partition sizes is in l^p-1 for some p \ge 1. Secondly, we have a partial converse which only applies to nonnegative functions, and if the ratio between the lengths smallest and biggest intervals in each partition is bounded uniformly. This says that if for some p \ge 1 f is not in L^p and the partition sizes are not in l^p-1, then the sequence of Riemann sums diverges with probability 1.

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