Basic Properties of a Measure

Let $(X,\mathcal{F},\mu)$ be a measure space, with the conventions for $[0,\infty]$ and for the sum of a sequence in $[0,\infty]$ fixed in that definition; in particular $\mathcal{F}$ is a $\sigma$-algebra on $X$.

1. (Finite additivity)¶ Let $r\in\mathbb{N}$ and let $A_1,\dots,A_r\in\mathcal{F}$ be pairwise disjoint. Then

$$\mu\Bigl(\bigcup_{i=1}^{r}A_i\Bigr)=\sum_{i=1}^{r}\mu(A_i),$$

the right-hand side being formed in $[0,\infty]$ and equal to $\infty$ exactly when $\mu(A_i)=\infty$ for some $i$.

2. (Monotonicity)¶ If $A,B\in\mathcal{F}$ and $A\subseteq B$, then $\mu(A)\le\mu(B)$.

3. (Differences)¶ If $A,B\in\mathcal{F}$, $A\subseteq B$ and $\mu(B)<\infty$, then $\mu(A)$ and $\mu(B\setminus A)$ are real and

$$\mu(B\setminus A)=\mu(B)-\mu(A).$$

In particular, if $\mu$ is finite then $\mu(X\setminus A)=\mu(X)-\mu(A)$ for every $A\in\mathcal{F}$.

4. (Countable subadditivity)¶ For every sequence $(A_m)_{m\in\mathbb{N}}$ in $\mathcal{F}$,

$$\mu\Bigl(\bigcup_{m\in\mathbb{N}}A_m\Bigr)\le\sum_{m\in\mathbb{N}}\mu(A_m).$$

5. (Continuity from below)¶ Let $(A_m)_{m\in\mathbb{N}}$ be a sequence in $\mathcal{F}$ with $A_m\subseteq A_{m+1}$ for every $m\in\mathbb{N}$, and let $A=\bigcup_{m\in\mathbb{N}}A_m$. If every $\mu(A_m)$ is real and the set $\{\mu(A_m):m\in\mathbb{N}\}$ is bounded above, then $\mu(A)$ is the least upper bound of that set and the sequence $(\mu(A_m))_{m\in\mathbb{N}}$ converges to $\mu(A)$. Otherwise $\mu(A)=\infty$.