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{\bf Question}
Let $E$ be a bounded measurable subset of the plane. Let $H(t)$
be the half-plane defined by
$$H(t)=\{(x,y)|x\leq t\}.$$
The function $f(t)$ is defined for all real $t$ by
$$f(t)=m(E\cap H(t)),$$
where $m$ denotes Lebesgue measure in the plane.
${}$
Prove that $f$ is a continuous function, and deduce that there is
a line in the plane parallel to the y-axis which bisects $E$ in
the sense that a subset of $E$ of measure $\frac{1}{2}m(E)$ lies
on each side of the line.
${}$
Give an example of a set $E$ for which the set
$f^{-1}(\frac{1}{2}m(E))$ contains more than one point.
${}$
Show that for all sets $E$, either $f^{-1}(\frac{1}{2}m(E))$ is a
single point or it is a closed interval.
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{\bf Answer}
$E$ is bounded and so $E$ is a subset of some square S with sides
of length $l$ parallel to the co-ordinate axes.
If $t_1