I'm studying from Ambrosio-Tilli "Topics on analysis metric spaces" and i think there is an errata corrige. Can someone help me to find the errata corrige book online if there is? However, maybe it is correct and i'm not understanding.
Let $\mu$ be an outer measure and let $\{A_i\}$ be a sequence of measurable sets. Then the following statements hold.
- If $A_i$s are pairwise disjoint, then $$\mu\left(\bigcup_i A_i \right) = \sum_i \mu(A_i) $$
- If $A_1 \subseteq \dots A_i \subseteq A_{i+1} \dots$, then $$ \lim_{i \to \infty} \mu (A_i) = \mu \left( \bigcup_i A_i \right)$$
- If $A_1 \supseteq \dots A_i \supseteq A_{i+1} \dots$, then $$ \lim_{i \to \infty} \mu (A_i) = \mu \left( \bigcap_i A_i \right)$$
I understand the proof of the first two statements. For the third we define the sets $B:= \bigcap_j A_j$ and $B_j := A_j \setminus A_{j+1}$ and observe that $B$ and $B_j$s are pairwise disjoint.
Now, in the book is written "{using the second statement}" we obtain:
$$ \mu(A_1) = \mu(B) + \lim_{n \to \infty} \mu \left( \bigcup_{j=1}^{n-1} B_j \right) = \mu(B) + \lim_{n \to \infty} (\mu(A_1)-\mu(A_n)) $$
But this equivalences is not for the second statement but just for the first one. Am i right?
Furthemore, the second statement is still true if $\mu$ is regular and the $A_i$s are not necessarily measurable. For prove this last statement we know that exist $\hat A_i$ containing them and such that $\mu(\hat A_i)=\mu(A_i)$ and we can do the same with their union $A_\infty$. But, given that it's not necessarily true that $\hat A_i \subseteq \hat A_{i+1}$ for all i, I have to define a sequence of set with this property. Define $$\tilde A_i:= \hat A_\infty \cap \bigcap_{j \geq i} \hat A_j \subseteq \hat A_i$$.
Is correct to say that: $$ \tilde A_i = \hat A_\infty \cap \bigcap_{j \geq i} \hat A_j \supseteq A_\infty \cap \bigcap_{j \geq i} A_j = A_i ?$$
In this way i prove that $\mu( \tilde A_i) = \mu (A_i)$. So i can conclude that $\tilde A_\infty$ is a measurable set that contain $A_\infty$ and proved the last statemend using the second statement to $\tilde A_i$