# Set Functions Examples 1

Recall from the Set Functions page that if $\mathcal A \subseteq \mathcal P(\mathbb{R})$ then a set function on $\mathcal A$ is a function $m : \mathcal A \to [0, \infty) \cup \{ \infty \}$.

- We said that a set function defined on $\mathcal P(\mathbb{R})$ is translation invariant if for all $y \in \mathbb{R}$ we have that:

- We said that a set function on $\mathcal P(\mathbb{R})$ is countably additive if for every countable collection of sets $\{ E_k \}_{k=1}^{\infty}$ we have that:

- And countably subadditive if:

The definitions of translation invariance and countable additivity can also be defined when $\mathcal A$ is a $\sigma$-algebra.

## Example 1

**Prove that if $\mathcal A$ is a $\sigma$-algebra, $m : \mathcal A \to [0, \infty) \cup \{ \infty \}$ is a set function with the countable additivity property, and $A, B \in \mathcal A$ are such that $A \subseteq B$, then $m(A) \leq m(B)$.**

Let $A \subseteq B$. Then the set $B \setminus A = B \cap A^c \in \mathcal A$ since $\mathcal A$ is a $\sigma$-algebra. Consider the collection $\{ A, B \setminus A \}$. This is a countable collection of sets from $\mathcal A$. Since $m$ has the countable additivity property we have that:

(4)## Example 2

**Prove that if $\mathcal A$ is a $\sigma$-algebra, $m : \mathcal A \to [0, \infty) \cup \{ \infty \}$ is a set function with the countable additivity property, and if there exists a set $A \in \mathcal A$ with $m(A) < \infty$ then $m(\emptyset) = 0$.**

$\{ \emptyset, A \}$ is a countable collection of sets from $\mathcal A$. So by the countable additivity property of $m$ we have that:

(5)Therefore $m(\emptyset) = 0$.