**Area** is a **set function** mapping from a class of so-called **measurable** sets $\mathscr{M}$ into the real numbers.
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Basic
What is a set function?
Back: A function mapping a collection of sets to real numbers.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the first set function Apostol introduces?
Back: Area.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What kind of mathematical entity is area?
Back: A function.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the domain of the area function?
Back: The class of measurable sets.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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What is the codomain of the area function?
Back: The real numbers.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the "function signature" of the area function $a$?
Back: $a \colon \mathscr{M} \rightarrow \mathbb{R}$ where $\mathscr{M}$ is the class of measurable sets.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What does Apostol mean by a measurable set?
Back: A set that can be ascribed an area.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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## Axioms
We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and a set function $a$, whose domain is $\mathscr{M}$, with the following six properties:
### Nonnegative Property
For each $S \in \mathscr{M}$, $a(S) \geq 0$.
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What does the nonnegative property of area state?
Back: For every set $S \in \mathscr{M}$, $a(S) \geq 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
State the nonnegative property of area in FOL.
Back: $\forall S \in \mathscr{M}, a(S) \geq 0$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S \in \mathscr{M}$. Why can't $a(S) = -1$?
Back: This violates the nonnegative property of $a$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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### Additive Property
If $S, T \in \mathscr{M}$, then $S \cup T$ and $S \cap T$ are in $\mathscr{M}$. Also $$a(S \cup T) = a(S) + a(T) - a(S \cap T).$$
Notice this last formulation is a special case of [[inclusion-exclusion|PIE]].
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Basic
Suppose $S, T \in \mathscr{M}$. What set(s) does the additive property of area state are also in $\mathscr{M}$?
Back: $S \cup T$ and $S \cap T$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cup T \in \mathscr{M}$?
Back: The additive property of $a$ states it is.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S, T \in \mathscr{M}$. Why is $S \cap T \in \mathscr{M}$?
Back: The additive property of $a$ states it is.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S, T \in \mathscr{M}$. What does $a(S \cup T)$ evaluate to?
Back: $a(S) + a(T) - a(S \cap T)$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
The additive property of area uses what combinatorial concept?
Back: The principle of inclusion/exclusion.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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### Difference Property
If $S, T \in \mathscr{M}$ such that $S \subseteq T$, then $T - S \in \mathscr{M}$ and $$a(T - S) = a(T) - a(S).$$
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Suppose $S, T \in \mathscr{M}$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
Back: N/A.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
Suppose $S, T \in \mathscr{M}$ such that $S \subseteq T$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
Back: $T - S$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $S, T \in \mathscr{M}$ such that $T \subseteq S$. What set(s) does the difference property of area state are also in $\mathscr{M}$?
Back: $S - T$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. Why is $T - S \in \mathscr{M}$?
Back: The difference property of $a$ states it is.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $a$ is an area function and $S, T \in \mathscr{M}$ s.t. $S \subseteq T$. What does $a(T - S)$ evaluate to?
Back: $a(T) - a(S)$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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### Invariance Under Congruence
If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$.
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What does the invariance of congruence property of area state?
Back: If $S \in \mathscr{M}$ and $T$ is congruent to $S$, then $T \in \mathscr{M}$ and $a(S) = a(T)$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What set(s) does the invariance of congruence property of area state are also in $\mathscr{M}$?
Back: $T$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $S \in \mathscr{M}$ and $T$ is congruent to $S$. What does $a(T)$ evaluate to?
Back: $a(S)$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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### Choice of Scale
Every rectangle $R$ is in $\mathscr{M}$. If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
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Basic
What shape is the choice of scale property of area concerned with?
Back: Rectangles.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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What sets does the choice of scale property of area state are also in $\mathscr{M}$?
Back: All rectangles.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $R$ is a rectangle. What property of area claims $R$ is measurable?
Back: Choice of scale.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Suppose $R$ is a rectangle. What does $a(R)$ evaluate to?
Back: If $R$ has edges of length $h$ and $k$, $a(R) = hk$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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What is the area of a line segment?
Back: $0$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
The line segment is considered a special case of what other shape?
Back: A rectangle.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
How does a rectangle relate to a line segment?
Back: A line segment is a rectangle with one dimension equal to zero.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
What is the area of a point?
Back: $0$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
The point is considered a special case of what other shape?
Back: A rectangle.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Basic
How does a rectangle relate to a point?
Back: A point is a rectangle with both dimensions equal to zero.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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### Exhaustion Property
Let $Q$ be a set. If there exists exactly one $c$ such that $a(S) \leq c \leq a(T)$ for all step regions $S$ and $T$ satisfying $S \subseteq Q \subseteq T$, then $Q \in \mathscr{M}$ and $a(Q) = c$.
Let $Q$ be a set. The {exhaustion} property of area states:
if there exists {exactly one} $c \in \mathbb{R}$ such that {$a(S) \leq c \leq a(T)$} for all {step regions} $S$ and $T$ satisfying {$S \subseteq Q \subseteq T$}, then {$Q \in \mathscr{M}$} and {$a(Q) = c$}.
