Generalized Lebesgue points for Sobolev functions
Karak, N. (2017). Generalized Lebesgue points for Sobolev functions. Czechoslovak Mathematical Journal, 67 (1), 143-150. doi:10.21136/CMJ.2017.0405-15
Published inCzechoslovak Mathematical Journal
© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017. Published in this repository with the kind permission of the publisher.
In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ Ms,p(X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of Hh-Hausdorff measure zero for a suitable gauge function h.