Theorem: Outer Jordan Content of Scaled Set

Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.
Let $c_1, c_2, \dotsc, c_n \in \R_{\ge 0}$ be non-negative real numbers.
Let $M' \subseteq \R^n$ be defined as:
:$M' = \set {\tuple {c_1 x_1, c_2 x_2, \dotsc, c_n x_n} : \tuple {x_1, x_2, \dotsc, x_n} \in \R^n}$
Then:
:$\map {m^*} {M'} = c_1 c_2 \dotsm c_n \map {m^*} M$
where $m^*$ denotes the outer Jordan content.
{{explain|Include somewhere in this page a link to a definition of "scaled set"}}

Proof:
Let $\epsilon > 0$ be arbitrary.
By Characterizing Property of Infimum of Subset of Real Numbers, let $C$ be a finite covering of $M$ by closed $n$-rectangles such that:
:$\ds \sum_{R \mathop \in C} \map V C < \map {m^*} M + \frac \epsilon {c_1 c_2 \dotsm c_n + 1}$
Let $C'$ be defined as:
:$\ds C' = \set {\closedint {c_1 a_1} {c_1 b_1} \times \dotso \times \closedint {c_n a_n} {c_n b_n} : \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n} \in C}$
For any $\tuple {c_1 x_1, \dotsc, c_n x_n} \in M'$, there must be some:
:$R = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n} \in C$
such that:
:$\tuple {x_1, \dotsc, x_n} \in R$
by definition of a covering.
That is:
{{begin-eqn}}
{{eqn | l = a_1
      | o = \le
      | m = x_1
      | ro = \le
      | r = b_1
}}
{{eqn | l = a_2
      | o = \le
      | m = x_2
      | ro = \le
      | r = b_2
}}
{{eqn | o =
      | m = \vdots
}}
{{eqn | l = a_n
      | o = \le
      | m = x_n
      | ro = \le
      | r = b_n
}}
{{end-eqn}}
Therefore, by Real Number Ordering is Compatible with Multiplication:
{{begin-eqn}}
{{eqn | l = c_1 a_1
      | o = \le
      | m = c_1 x_1
      | ro = \le
      | r = c_1 b_1
}}
{{eqn | l = c_2 a_2
      | o = \le
      | m = c_2 x_2
      | ro = \le
      | r = c_2 b_2
}}
{{eqn | o =
      | m = \vdots
}}
{{eqn | l = c_n a_n
      | o = \le
      | m = c_n x_n
      | ro = \le
      | r = c_n b_n
}}
{{end-eqn}}
Or, in other words:
:$\tuple {c_1 x_1, \dotsc, c_n x_n} \in \closedint {c_1 a_1} {c_1 b_1} \times \dotso \times \closedint {c_n a_n} {c_n b_n} \in C'$
Hence:
:$C'$ is a finite covering of $M'$ by closed $n$-rectangles.
Therefore:
{{begin-eqn}}
{{eqn | l = \map {m^*} {M'}
      | o = \le
      | r = \sum_{R \mathop \in C'} \map V R
      | c = {{Defof|Outer Jordan Content}}
}}
{{eqn | r = \sum_{\sqbrk {\closedint {\mathbf a} {\mathbf b} } \mathop \in C} \prod_{i \mathop = 1}^n \paren {c_i b_i - c_i a_i}
      | c = $\map V R$ as in {{Defof|Outer Jordan Content}}
}}
{{eqn | r = \sum_{\sqbrk {\closedint {\mathbf a} {\mathbf b} } \mathop \in C} \prod_{i \mathop = 1}^n c_i \paren {b_i - a_i}
}}
{{eqn | r = \sum_{\sqbrk {\closedint {\mathbf a} {\mathbf b} } \mathop \in C} c_1 c_2 \dotsm c_n \prod_{i \mathop = 1}^n \paren {b_i - a_i}
}}
{{eqn | r = c_1 c_2 \dotsm c_n \sum_{R \mathop \in C} \map V C
      | c = $\map V R$ as in {{Defof|Outer Jordan Content}}
}}
{{eqn | o = \le
      | r = c_1 c_2 \dotsm c_n \paren {\map {m^*} M + \frac \epsilon {c_1 c_2 \dotsm c_n + 1} }
      | c = By construction of $C$
}}
{{eqn | r = c_1 c_2 \dotsm c_n \map {m^*} M + \frac {c_1 c_2 \dotsm c_n} {c_1 c_2 \dotsm c_n + 1} \epsilon
}}
{{eqn | o = <
      | r = c_1 c_2 \dotsm c_n \map {m^*} M + \epsilon
}}
{{end-eqn}}
As $\epsilon > 0$ was arbitrary, it follows from Real Plus Epsilon that:
:$\map {m^*} {M'} \le c_1 c_2 \dotsm c_n \map {m^*} M$
{{qed|lemma}}
Now, if every $c_i > 0$, we have that:
:$M = \set {\tuple {\frac 1 {c_1} x_1, \dotsc, \frac 1 {c_n} x_n} : \tuple {x_1, \dotsc, x_n} \in M'}$
Therefore, the above argument can be applied with the roles of $M$ and $M'$ interchanged to prove that:
:$\map {m^*} M \le \frac 1 {c_1 c_2 \dotsm c_n} \map {m^*} {M'}$
which is to say:
:$c_1 c_2 \dotsm c_n \map {m^*} M \le \map {m^*} {M'}$
If the above does not hold, then $c_i = 0$ for some $i \in \set {1, 2, \dotsc, n}$.
Then:
:$c_1 c_2 \dotsm c_n \map {m^*} M = 0 \le \map {m^*} {M'}$
In either case, we have:
:$c_1 c_2 \dotsm c_n \map {m^*} M \le \map {m^*} {M'}$
The result follows from the two inequalities.
{{qed}}
Category:Outer Jordan Content
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