Why Is a Rectangle Considered Closed and Bounded in Volume Proofs?

In summary, the conversation discusses the proof of the theorem that for any rectangle, the outer measure is equal to the volume. The speaker wonders why it is only necessary to consider the case where the rectangle is closed and bounded, and suggests that it may have to do with how open and closed rectangles can be written as a union of closed intervals. They also question if there may be another reason for this assumption in the proof.
  • #1
mathmari
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Hey! :eek:

I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume.

At the beginning of the proof there is the following sentence:

It is enough to look at the case where the rectangle R is closed and bounded.

Why does it stand?? (Wondering)

Is it maybe as followed??A closed rectangle is
$$[a_1,b_1] \times [a_2,b_2] \times \dots \times [a_d,b_d]$$

An open rectangle is
$$(a_1,b_1) \times (a_2,b_2) \times \dots \times (a_d,b_d)$$
which can be written as a union of closed intervals.
 
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  • #2
mathmari said:
Is it maybe as followed??A closed rectangle is
$$[a_1,b_1] \times [a_2,b_2] \times \dots \times [a_d,b_d]$$

An open rectangle is
$$(a_1,b_1) \times (a_2,b_2) \times \dots \times (a_d,b_d)$$
which can be written as a union of closed intervals.

Or is there an other reason, why it is sufficient to suppose that $R$ is closed and bounded to prove that $m^*(R)=v(R)$?? (Wondering)
 

FAQ: Why Is a Rectangle Considered Closed and Bounded in Volume Proofs?

What does it mean for a set to be "closed and bounded" in R?

In the context of the real numbers (R), a set is considered closed if it contains all of its boundary points. It is considered bounded if there exists some number M such that all elements of the set have an absolute value less than or equal to M. Therefore, a set is "closed and bounded" in R if it includes all of its endpoints and is limited in size.

How does the concept of "closed and bounded" relate to continuity in R?

In R, a function is continuous on a closed and bounded set if and only if it is bounded and achieves its maximum and minimum values at the endpoints of the set. This is known as the Extreme Value Theorem. Therefore, the concept of "closed and bounded" is important in determining the continuity of a function in R.

Can a set be closed but not bounded in R?

Yes, it is possible for a set to be closed but not bounded in R. An example of this is the set of all real numbers greater than or equal to 0, which includes its endpoints (0) but is not limited in size.

Is a closed and bounded set in R always compact?

Yes, a closed and bounded set in R is always compact. This is known as the Heine-Borel Theorem, which states that a subset of R is compact if and only if it is closed and bounded.

How do you prove that a set is closed and bounded in R?

To prove that a set is closed and bounded in R, you must show that it contains all of its boundary points and that it is limited in size. This can be done using mathematical proofs and definitions. Additionally, you can use the Heine-Borel Theorem to show that a set is compact, which implies that it is closed and bounded.

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