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I am trying to solve an introductory exercise in measure theory, but I am stuck.

$X=\{r:r\in\mathbb{Q}, r\in[0,1]\}$, $\mathbb{A}$ is an algebra made up of finite unions of the following of sets: $$1) \{r:a<r<b\}$$$$2) \{r:a<r\leq b\}$$$$3) \{r:a\leq r<b\}$$$$4) \{r:a\leq r\leq b\}$$ and define a set function $p(A)=b-a$ for $A$ as above. Show that p is finitely additive measure.

a, b are not specificed, but I think could be assumed to be in $\mathbb{R}$. and $r\in\mathbb{Q}$ still holds for these 4 sets.

It's simple to show that for disjoint $A_1,A_2\in \mathbb{A}$ when they combine such that $A_1+A_2$ becomes of the type mentioned above.

But I'm not sure how to show that for $A_1,A_2\in \mathbb{A}$, when there exists an nonempty set between $a_1$ and $b_2$ that doesn't belong to the union.

Example of where I'm stuck in case of confusion: $$p(\{r:a_1\leq r\leq b_1\})+p(\{r:a_2\leq r\leq b_2\})=p(\{r:a_1\leq r\leq b_1 \space or\space a_2\leq r\leq b_2\})$$ for $b_1<a_2$.

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  • $\begingroup$ Perhaps you need to define $p$ on more sets: finite disjoint unions of sets of the types 1...4. In order to make the domain of $p$ an algebra. $\endgroup$
    – GEdgar
    Commented Nov 8, 2022 at 18:36
  • $\begingroup$ Perhaps I expressed it poorly, but the algebra defined in this exercise is indeed finite unions of the sets of type 1 to 4. I will edit the post itself for this. $\endgroup$
    – Waaal
    Commented Nov 8, 2022 at 19:03
  • $\begingroup$ I am sure that by "algebra", they mean with respect to the operations of union and intersection of sets, not addition - in the vein of "sigma algebra", but with only finite unions and intersections. And I expect the intent is to extend $p$ from rational intervals to all of $\Bbb A$ by the rule that if $U \in \Bbb A$ and $U = I_1 \cup I_2 \cup \dots \cup I_n$, where the $I_i$ are disjoint intervals, then $p(U) := p(I_1) + \dots + p(I_n)$. $\endgroup$ Commented Nov 9, 2022 at 14:13

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