# Questions about Attachments: Outer Measure and Infinity

• Artusartos
In summary, the author is confused about why the measure of E is not equal to the measure of [O_k \sim E_k]. They are also confused about how E can have an open cover when it is equal to the real line.
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?

#### Attachments

• Rodyen40.jpg
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• Royden41.jpg
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Last edited:
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}

joeblow said:
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 :)

## 1. What is outer measure and how is it different from standard measure?

Outer measure is a mathematical concept that measures the size or length of a set. It differs from standard measure because it can be applied to non-measurable sets, such as fractals or irrational numbers.

## 2. How is outer measure calculated?

Outer measure is calculated by taking the infimum (greatest lower bound) of the sum of the lengths of all intervals that cover the set. In simpler terms, it is the smallest possible sum of intervals that can cover the set.

## 3. Can outer measure be infinite?

Yes, outer measure can be infinite. This can happen when the set being measured is unbounded, meaning it has no upper or lower limit. For example, the set of all real numbers has an infinite outer measure.

## 4. What is the relationship between outer measure and infinity?

Outer measure is closely related to the concept of infinity. In fact, one of the main uses of outer measure is to determine the size or length of sets that are too large or infinite to be measured by standard means.

## 5. How is outer measure used in real-world applications?

Outer measure has various applications in fields such as mathematics, physics, and engineering. It is used to measure the size and length of complex objects, such as fractals and irregular shapes, and to calculate probabilities in probability theory. It is also used in the study of infinite sets and their properties.

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