Construct measures on $\sigma(B)$ that agree on $B$

Question

Let $X=\{ 1,2,3,4\}$ and $\mathcal B=\{\{1,2 \},\{ 1,3\},\{ 2,4\},\{ 3,4\} \}$. And let $\mathscr A = \sigma(\mathcal B)$ be the $\sigma$-algebra generated by the set $\mathcal B$. I wish to construct two different measures that agree on $\mathcal B$.

I try to write down some measures like: $A \in \mathscr A$

• $m(A) = $ number of elements in $A$

• $m(A) = $ average on $A$

• $m(A) = $ sum of $A$

And more, however none of them agree on $\mathcal B$.

Answer 1

Define $\mu\{1\} =\mu\{4\}=1$, $\mu\{2\}=\mu\{3\}=0$ and $\mu\{1\} =\mu\{4\}=0$, $\mu\{2\}=\mu\{3\}=1$. Then

• $\mu\{1,2\}=1=\nu\{1,2\}$;
• $\mu\{1,3\}=1=\nu\{1,3\}$;
• $\mu\{2,4\}=1=\nu\{2,4\}$ and
• $\mu\{3,4\}=1=\nu\{3,4\}$,

but $\mu$ and $\nu$ are far from agreeing on the $\sigma$-algebra $\mathcal A$.