Inner Jordan Content is Well-Defined

Theorem

Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.

Let:

$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$

$\ds R' = \prod_{i \mathop = 1}^n \closedint {a'_i} {b'_i}$

be closed $n$-rectangles that contain $M$.

Let $V_R, V_{R'} \in \R_{\ge 0}$ be defined as:

$\ds V_R = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$

$\ds V_{R'} = \prod_{i \mathop = 1}^n \paren {b'_i - a'_i}$

Then:

$V_R - \map {m^*} {R \setminus M} = V_{R'} - \map {m^*} {R' \setminus M}$

Therefore, the inner Jordan content of $M$ is independent of the rectangle $R$ used to define it.

Proof

For each $k \in \set {0, 1, \dotsc, n}$, define:

$\ds R_k = \prod_{i \mathop = 1}^k \closedint {a_i} {b_i} \times \prod_{i \mathop = k + 1}^n \closedint {a'_i} {b'_i}$

$\ds V_k = \prod_{i \mathop = 1}^k \paren {b_i - a_i} \times \prod_{i \mathop = k + 1}^n \paren {b'_i - a'_i}$

In particular, we have:

$R_0 = R$

$R_n = R'$

$V_0 = V_R$

$V_n = V_{R'}$

We will prove that, for every $k \in \set {1, 2, \dotsc, n}$:

$\paren 1 \quad V_{k - 1} - \map {m^*} {R_{k - 1} \setminus M} = V_k - \map {m^*} {R_k \setminus M}$

From this, we would have:

\(\ds V_R - \map {m^*} {R \setminus M}\)

\(=\)

\(\ds V_0 - \map {m^*} {R_0 \setminus M}\)

Definition of $R_0$

\(\ds \)

\(=\)

\(\ds V_1 - \map {m^*} {R_1 \setminus M}\)

By $\paren 1$, with $k = 1$

\(\ds \)

\(=\)

\(\ds V_2 - \map {m^*} {R_2 \setminus M}\)

By $\paren 1$, with $k = 2$

\(\ds \)

\(\vdots\)

\(\ds \)

\(\ds \)

\(=\)

\(\ds V_n - \map {m^*} {R_n \setminus M}\)

By $\paren 1$, with $k = n$

\(\ds \)

\(=\)

\(\ds V_{R'} - \map {m^*} {R' \setminus M}\)

Definition of $R_n$

which is the stated result.

Let $k \in \set {1, 2, \dotsc, n}$ be arbitrary.

We want to show that:

$V_{k - 1} - \map {m^*} {R_{k - 1} \setminus M} = V_k - \map {m^*} {R_k \setminus M}$

Or equivalently:

$\map {m^*} {R_k \setminus M} - \map {m^*} {R_{k - 1} \setminus M} = V_k - V_{k - 1}$

We can compute:

\(\ds V_k - V_{k - 1}\)

\(=\)

\(\ds \prod_{i \mathop = 1}^k \paren {b_i - a_i} \times \prod_{i \mathop = k + 1}^n \paren {b'_i - a'_i} - \prod_{i \mathop = 1}^{k - 1} \paren {b_i - a_i} \times \prod_{i \mathop = k}^n \paren {b'_i - a'_i}\)

Definition of $V_k$

\(\ds \)

\(=\)

\(\ds \prod_{i \mathop = 1}^{k - 1} \paren {b_i - a_i} \times \prod_{i \mathop = k + 1}^n \paren {b'_i - a'_i} \times \paren {\paren {b_k - a_k} - \paren {b'_k - a'_k} }\)

\(\ds \)

\(=\)

\(\ds \prod_{i \mathop = 1}^{k - 1} \paren {b_i - a_i} \times \prod_{i \mathop = k + 1}^n \paren {b'_i - a'_i} \times \paren {\paren {a'_k - a_k} + \paren {b_k - b'_k} }\)

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