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**Opalg** Think about the problem in an informal, geometric way. You have a set *E* with positive measure, and you want to find an interval *I *such that the proportion of *E* lying inside *I* is *c*, in the sense that $\displaystyle m(E\cap I) = cm(E)$.

One way of doing that is to take a point *a* that moves along the real line from –∞ to +∞, and to look at the proportion of *E* that lies to the left of *a*. Call this proportion *f(a)*, so that $\displaystyle f(a)=m((-\infty,a)\cap E)/m(E)$. When *a* is very large and negative, practically none of E will lie to the left of a, so *f(a)* will be very small. As *a* increases, so does *f(a)*, and as *a*→+∞, *f(a)*→1. Also, *f(a)* is a continuous function of *a*, so by the intermediate value theorem there will be some point along the line at which *f(a)*=*c*. For that value of *a*, $\displaystyle m((-\infty,a)\cap E) = cm(E)$.

That was my motivation for suggesting $\displaystyle I = (-\infty,a)$, where $\displaystyle a = \sup\{x\in\mathbb{R}:m\bigl((-\infty,x)\cap E\bigr) < cm(E)\}$. The value of *a* given by that definition is the sup of all the points for which *f(a)*<*c*. At that point, *f(a)* will be equal to *c*.