Exercise 1.2.12 (Monotonicity and subadditivity as a consequense of additivity)

Question

Let $\mathcal{L}(\mathbb{R}^d)$ be the set of all Lebesgue measurable sets on $\mathbb{R}^d$, and let $m:\mathcal{L}	o[0,+\infty]$ be the map that obeys empty set and countable additivity axioms. Then $m$ also obeys monotonicity and countable subadditivity.

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(oman*)]
\item 	extit{(monotonicity)}
ewline
Let $A,B: A \subseteq B$. Then, we have the disjoint union $B = A \cup (B/A)$. Therefore, by additivity $m(B) = m(A) + m(B/A) \geq m(A)$ (since $m(B/A) \geq 0$ by definition).

\item 	extit{(countable subadditivity)}
ewline
Let $E_1, E_2 \ldots$ be a countable sequence of sets in $\mathbb{R}^d$. Notice that we can write the union of these sets in a disjoint form
$$\bigcup_{n=1}^{\infty} E_n = \bigcup_{n=1}^{\infty} E_n \setminus \left(\bigcup_{i=1}^{n-1} E_i ight).$$
This is not hard to demonstrate.
\begin{itemize}
\item[$\supseteq$] Pick an arbitrary $x\in \bigcup_{n=1}^{\infty} E_n \setminus \left(\bigcup_{i=1}^{n-1} E_i ight)$. Then for some $n\in\mathbb{N}$ we have $x \in  E_n \setminus \left(\bigcup_{i=1}^{n-1} E_i ight) \subseteq E_n \subseteq \bigcup E_n$.
\item[$\subseteq$] Pick an arbitrary $x \in \bigcup_{n=1}^{\infty} E_n$. Then $x \in E_n$ for some $n\in\mathbb{N}$. Set $n_0 := \min\{n: x \in E_n\}$. We then have $n_0 < n$; it thus follows $x \in E_{n_0} \setminus \left(\bigcup_{i=1}^{n_0-1} E_i ight)$, and we are done.
\item[$\coprod$] For all $n,n' \in \mathbb{N}: n' > n$ we obviously have $\left[E_n \setminus \left(\bigcup_{i=1}^{n-1} E_i ight) ight]\bigcap \left[E_{n'} \setminus \left(\bigcup_{i=1}^{n'-1} E_i ight)ight] = \emptyset$.
\end{itemize}

Using this, we obtain
$$m\left(\bigcup_{n=1}^{\infty} E_n ight) = \sum_{n=1}^{\infty} m\left( E_n \setminus \left(\bigcup_{i=1}^{n-1} E_i ight)ight) \leq  \sum_{n=1}^{\infty} m\left(E_night).$$
\end{enumerate}

Exercise 1.2.12 (Monotonicity and subadditivity as a consequense of additivity)

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Exercise 1.2.12 (Monotonicity and subadditivity as a consequense of additivity)

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