How do we measure the ”size” of a set in IR? Let’s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its length, which is used frequently in differentiation and integration. For any bounded interval I (open, closed, half-open) with endpoints a and b (a ≤ b), the length of I is defined by `(I) = b − a. Of course, the length of any unbounded...

Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measur...

in this work are studied the jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

Some new inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral are obtained. Applications for $f$-divergence measure are provided as well.