О метрическом свойст ве Дарбу

Abstract

A measurable functionf on [a, b] is called to possess the metric Darboux property if none of the intervalsI⊂[a, b] can be split into two measurable subsetsI′ andI″, of positive measure, such that

$$\mathop {\sup vrai}\limits_{x \in I'} f\left( x \right)< \mathop {\inf vrai}\limits_{x \in I''} f\left( x \right)$$

It is proved that each functionfεL _p [0, 1] for which

$$\mathop {\lim \inf }\limits_{\delta \to + 0} \delta ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} p}} \right. \kern-\nulldelimiterspace} p}} \omega _p \left( {f; \delta } \right)< \infty ,$$

can be represented as a sum of a function of boundedp-variation and a function possessing the metric Darboux property. It is proved that the above condition on the modulus of continuity cannot be weakened.

Certain problems connected with the lower symmetric densities of measurable sets are also considered.