Exercise 1.1.19 (Caratheodory type property)

Question

Let $E\subset \mathbb{R}^d$ be a bounded set, and let $F\subset \mathbb{R}^d$ be an elementary set. Show that
$$m^{*,(J)}(E) = m^{*,(J)}(E \cap F) + m^{*,(J)}(E/F)$$

Elu Thingol · Accepted Answer

We must demonstrate that

\begin{equation*}
\inf \left\{ m(B) : \begin{array}{l} E\subset B \ 	ext{$B$ is elementary}\end{array}    ight\} 
=
\inf \left\{ m(C) : \begin{array}{l} E \cap F \subset C \ 	ext{$C$ is elementary}\end{array}    ight\} + \inf \left\{ m(D) :      \begin{array}{l} E/ F \subset D \ 	ext{$D$ is elementary}\end{array}    ight\}
\end{equation*}

We demonstrate that the left-hand side is both greater than or equal and less than or equal to the right-hand side.

\begin{itemize}

\item[$\leq$] Let $C$ be an elementary outer cover of $E \cap F$, and let $D$ be an elementary outer cover of $E / F$, and consider the quantity $m(C) + m(D)$. Notice that $E = (E \cap F) \cup (E / F) \subseteq C \cup D$; in other words, $C \cup D$ is an elementary cover of $E$. By finite subadditivity of elementary measure we have $m(C \cup D) \leq m(C) + m(D)$, which proves the first part of the assertion.

\item[$\geq$] Let $B$ be an elementary outer cover of $E$. Then, $E \cap F \subseteq B \cap F$, and $E / F \subseteq B / F$. In other words, $B \cap F$ is an elementary outer cover of $E \cap F$, and $B / F$ is an elementary outer cover of $E / F$. Notice that we have by additivity property of elementary measure $m(B) = m((B \cap F) \cup (B / F)) = m(B\cap F) + m(B/F)$; thus, the left-hand side cannot exceed the right-hand side since we can always at least perform the above duplication.

\end{itemize}

Exercise 1.1.19 (Caratheodory type property)

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Exercise 1.1.19 (Caratheodory type property)

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