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

.

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

. 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*,

.

That was my motivation for suggesting

, where

. 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*.