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Artusartos

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I have 2 questions about the attachments.

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

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

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

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

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