Let *R* be any (bounded) region in the plane which we want to measure
the area of. We can tile the plane with squares of unit length and
count the number of such squares that are contained in the region to
obtain an approximation to the area from below. On the other hand we
can count the number of squares that meet to region to obtain an
approximation to the area from above. We can repeat this with squares
of smaller size and appropriately scale the count it seems clear that
the approximant from below will increase and the approximant from
above will decrease. The least upper bound of the former is called the
inner measure and the greatest lower bound of the latter the outer
measure. To obtain an area for the region we must show that these two
numbers are the same; moreover, we would like these numbers to be
independent of the placement of the grid as well as rotation and/or
shearing of the grid.

- Show that the sum
*L*(*P*,*f*) (respectively*U*(*P*,*f*)) approximate the area of the region from below (respectively above), i. e. are sums of areas of rectangles enclosed by (respectively enclosing) the region*R*.

*L*(*P*,*f*)= *m*_{i}(*t*_{i}-*t*_{i - 1})*U*(*P*,*f*)= *M*_{i}(*t*_{i}-*t*_{i - 1})

- If
*P'*is a finer partition than*P*(i. e. each point of*P*is also a point of*P'*) then show that*L*(*P*,*f*)*L*(*P'*,*f*)*U*(*P'*,*f*)*U*(*P*,*f*) - Let
*P*_{n}denote the partition of [*a*,*b*] into*n*equal parts. Show thatsup{Similarly for the infimum of the*L*(*P*,*f*)|*P*a partition } sup{*L*(*P*_{n},*f*)}*U*(*P*,*f*),inf{*U*(*P*,*f*)|*P*a partition } inf{*U*(*P*_{n},*f*)} - Let
*i*=*i*(*P*,*f*) be such that the difference*M*_{i}-*m*_{i}is maximum. Then show that*U*(*P*,*f*)-*L*(*P*,*f*) (*M*_{i}-*m*_{i})(*b*-*a*)*x*(*P*,*f*) denote the mid point of the interval [*i*_{i - 1},*t*_{i}] for this*i*. - Let
*c*be any point of the interval [*a*,*b*]. For any positive , show that there is a > 0 so that the difference between the maximum and minimum values of*f*(*x*) on the interval [*c*- ,*c*+ ] is less than /(*b*-*a*). (Hint: use continuity of*f*at*c*). - The sequence
{
*x*_{n}=*x*(*P*_{n},*f*)} has a convergent subsequence {*y*_{k}=*x*_{nk}}, with limit point*c*. Show that there is a*k*_{0}so that if*k**k*_{0}and*i*=*i*(*P*_{nk},*f*) then the entire sub-interval [*t*_{i - 1},*t*_{i}] of the partition*P*_{nk}is contained in [*c*- ,*c*+ ]. - Deduce that
sup{
*L*(*P*_{nk},*f*)} = inf{*U*(*P*_{nk},*f*)}. - Conclude that the inner and outer measure of the region
*R*coincide.

(*d*^{ . }*f* + *g*) = *d*^{ . }*f* + *g*

Now if *f* is continuous on an interval [*a*, *b*] where its minimum
value is *m* and its maximum value is *M* it is clear that