Their first general formulation is as follows : for any suitably restricted real function f \ colon S ^ n \ times X \ to \ mathbb { R }, there is a point of the-sphere such that the surface, dividing into and, simultaneously bisects the outer measure of.
32.
The concepts of " Dimension and outer measure " have experienced applications and further developments in many areas such as in the theory of dynamical systems, geometric measure theory, the theory of self-similar sets and fractals, the theory of stochastic processes, harmonic analysis, potential theory and number theory.
33.
Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of another set when E is used as a " mask " to " clip " that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure . ( Such sets are, in fact, not Lebesgue-measurable .)
