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.