We define a characteristic function, , on a set
of points on
by
Let be a step functions taking the (constant) value
on the
th open subinterval
of a partition of
. Prove that for each
, we have
Proof. First, we note that the open subintervals of some partition of are necessarily disjoint since
. Hence, if
then
for exactly one
.
So, we have
for all , and for any
. Further, by definition of
, we know
if
. So,
for each
I think the summation result should be c(j) instead of c(k) since you have already let x is an element of I(j)