On continuity of distribution function and decreasing rearrangement

Abstract

Given a measure space $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ , the distribution function ${\mu }_f(\nu )=\mu (\{t\in \mathrm{\Omega }:|f(t)|>\nu \})$ where $\nu ⩾0$ and the decreasing rearrangement $f^{*}(z)=inf\{\nu ⩾0:{\mu }_f(\nu )⩽z\}$ , where $z⩾0$ and by convention $inf\{\mathrm{\varnothing }\}=\mathrm{\infty }$ , of a measurable function $f$ are known to be right continuous functions. However, these functions need not be left continuous. The purpose of this paper is to investigate the conditions under which these functions are continuous. Under the assumption that $\mu (\{t\in \mathrm{\Omega }:|f(t)|>0\})<\mathrm{\infty }$ , we provide a necessary and sufficient condition for the function ${\mu }_f$ to be continuous at $\nu >0$ . Using the same we provide a similar result for the continuity of decreasing rearrangement $f^{*}$ of the function $f$ .