Questions about Attachments: Outer Measure and Infinity

[Artusartos]

I have 2 questions about the attachments.

1) In the second attachment, I'm a bit confused about the thing that I marked: $O \sim E = \cup^{\infty}_{k=1} O_k \sim E \subseteq \cup^{\infty}_{k=1} [O_k \sim E_k]$. I just don't understand how $\cup^{\infty}_{k=1} O_k \sim E$ can be smaller than $\cup^{\infty}_{k=1} [O_k \sim E_k]$. Isn't E equal to $\cup^{\infty}_{k=1} E_k$?

2) Also, they are considering the measure of E when it is equal to infinity. But since outer measure means length...it means that the length of E is infinity. Then how is it possible for E to have an open cover unless that open cover is equal to E itself. In other words, how can there be anything greater than E? But if it is equal, then wouldn't their difference be zero? So why do we even need to check if it is less than epsilon?

Thanks in advance

 

[joeblow]

1) Equality does not necessarily hold for infinite unions. Write $$O_k \sim E_k=O_k \cap \tilde{E_k}$$ and see that $$\cup [O_n \sim E_n]=(O_1 \cap \tilde{E_1}) \cup (O_2 \cap \tilde{E_2}) \cup \cdots$$ while $$\cup O_n \sim E = (O_1 \cup O_2 \cup \cdots)\cap \tilde{E}= (O_1 \cap \tilde{E})\cup (O_2 \cap \tilde{E}) \cup \cdots.$$ So they are not necessarily the same. Term-by-term, you can see that $$O_k \cap \tilde{E} \subseteq O_k \cap \tilde{E}_k$$
2) The real line contains R~{0} and both have the same measure. R is a cover of R~{0}

 

[Artusartos]

1) Equality does not necessarily hold for infinite unions. Write $$O_k \sim E_k=O_k \cap \tilde{E_k}$$ and see that $$\cup [O_n \sim E_n]=(O_1 \cap \tilde{E_1}) \cup (O_2 \cap \tilde{E_2}) \cup \cdots$$ while $$\cup O_n \sim E = (O_1 \cup O_2 \cup \cdots)\cap \tilde{E}= (O_1 \cap \tilde{E})\cup (O_2 \cap \tilde{E}) \cup \cdots.$$ So they are not necessarily the same. Term-by-term, you can see that $$O_k \cap \tilde{E} \subseteq O_k \cap \tilde{E}_k$$
2) The real line contains R~{0} and both have the same measure. R is a cover of R~{0}

Thank you so much :)