1. Definition of areas
Definition 1.1. The area of a rectangle R of length a and of width b is defined to be A(R) = ab.
A simple plane region is a union of finitely many rectangles. A simple plane region can be decom- posed into a finite union of non-overlapping rectangles. Two rectangles are called non-overlapping if their intersection is contained in their boundaries.
Definition 1.2. If S is a simple region and S = R1∪ · · · ∪ RN where Ri and Rj are nonoverlapping rectangles for all i 6= j, we define the area of S to be
A(S) = A(R1) + · · · + A(RN).
Let Ω be a bounded plane region. We define the inner area of Ω to be the real number A−(Ω) = sup{A(S) : S is a simple region, S ⊆ Ω}
and the outer area of Ω to be the real number
A+(Ω) = inf{A(S0) : S0 is a simple region, S0 ⊇ Ω}.
One can see that
A−(Ω) ≤ A+(Ω).
Definition 1.3. Let Ω be a bounded plane region. We say that Ω has an area if A+(Ω) = A−(Ω).
In this case, we call the number A+(Ω) = A−(Ω) the area of Ω denoted by A(Ω).
Example 1.1. Let Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x2}. Show that Ω has an area and find its area.
Let n be a natural number and for each 1 ≤ i ≤ n − 1, we consider the rectangle Ri =
(x, y) : i
n ≤ x ≤ i + 1
n , 0 ≤ y ≤ i2 n2
.
Then S = R1∪ · · · ∪ Rn−1 is a simple region contained in Ω. The area of Ri is given by i2/n3 and hence the area of S is
A(S) = A(R1) + · · · + A(Rn−1) =
n−1
X
i=1
i2
n3 = (n − 1)(2n − 1)
6n2 .
By definition, A(S) ≤ A−(Ω) and hence
(n − 1)(2n − 1)
6n2 ≤ A−(Ω).
Similarly, we consider the rectangles R0i=
(x, y) : i − 1
n ≤ x ≤ i
n, 0 ≤ y ≤ i2 n2
for each 1 ≤ i ≤ n. Then S0= R01∪ · · · ∪ R0n is a simple region containing Ω. Then, A(S0) = A(R01) + · · · + A(R0n) =
n
X
i=1
i3
n3 = (n + 1)(2n + 1) 6n2 . By definition A+(Ω) ≤ A(S0) and therefore
A+(Ω) ≤ (n + 1)(2n + 1)
6n2 .
Since A−(Ω) ≤ A+(Ω), we obtain that for each n ≥ 1, (n − 1)(2n − 1)
6n2 ≤ A−(Ω) ≤ A+(Ω) ≤ (n + 1)(2n + 1) 6n2 . By taking n → ∞,
1
3 ≤ A−(Ω) ≤ A+(Ω) ≤ 1 3.
1
2
We find that
A+(Ω) = A−(Ω) = 1 3. Hence we find that Ω has an area and its area equals to 1/3.
1.1. Appendix: Least upper bound and the greatest lower bound. Let K be a collection of real numbers.
Definition 1.4. We say that K is bounded above if there exists a real number U such that x ≤ U for all x ∈ K.
In this case, U is called an upper bound for K. The smallest upper bound for K denoted by sup K is called the least upper bound for K. (sup K is an upper bound for K and if U is an upper bound for K, then sup K ≤ U.)
Definition 1.5. We say that K is bounded below if there exists a real number L such that x ≥ L for all x ∈ K.
In this case, L is called a lower bound for K. The largest lower bound for K denoted by inf K is called the greatest lower bound for K. (inf K is a lower bound for K and if L is a lower bound for K, then inf K ≥ L.)
Example 1.2. Let K = (0, 1) the set of all real numbers x such that 0 < x < 1. 2 is an upper bound for K since x < 2 for all x ∈ K. 2 is not the least upper bound for K since 1 < 2 and 1 is an upper bound for K. In fact,
sup K = 1, inf K = 0.
