# Question about a measure of a set

• cragar
In summary: Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.In summary, the conversation discusses the possibility of using the fact that all countable sets have zero measure to prove the existence of a larger infinity. The speaker suggests using boxes and widths to show that the sum can be made arbitrarily small. However, they also mention that this proof relies on measure theory and suggest using the convergence of an infinite series instead. The conversation concludes with the confirmation that this approach is valid, but still requires some elementary facts from measure theory.
cragar
Could we use the fact that all countable sets have zero measure
to prove that their must be a larger infinity.
we know countable sets have measure zero because I could just start by making
boxes around each number and then add up their widths.
For the first number I will make a box that has width $\frac{\epsilon}{2}$
and then each box will have half the width of the previous box.
so the sum will be $\epsilon(1/2+1/4+1/8...)$
and i can make $\epsilon$ as small as I want.
This proof comes from Gregory Chaitin.
If the reals were countable they would have measure zero, but we know this isn't true
because the reals have positive width. Can i do this to prove there is a larger infninty than countable.

cragar said:
Could we use the fact that all countable sets have zero measure
to prove that their must be a larger infinity.
we know countable sets have measure zero because I could just start by making
boxes around each number and then add up their widths.
For the first number I will make a box that has width $\frac{\epsilon}{2}$
and then each box will have half the width of the previous box.
so the sum will be $\epsilon(1/2+1/4+1/8...)$
and i can make $\epsilon$ as small as I want.
This proof comes from Gregory Chaitin.
If the reals were countable they would have measure zero, but we know this isn't true
because the reals have positive width. Can i do this to prove there is a larger infninty than countable.

Short answer - yes. The only objection, compared to Cantor proof, is that it is necessary to develop measure theory first.

ok thanks for your answer. Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.

cragar said:
ok thanks for your answer. Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.
Yes - although if you look at it closely you will find you are using some elementary facts from measure theory.

ok thanks

## 1. What is a measure of a set?

A measure of a set is a mathematical concept that assigns a number to a set in order to determine its size or extent. It is often used in the study of measure theory and is closely related to the concepts of length, area, and volume.

## 2. How is a measure of a set calculated?

The calculation of a measure of a set depends on the specific measure being used. However, in general, a measure of a set is calculated by determining the size or extent of the set in relation to a chosen unit of measurement. This can involve techniques such as integration, counting, or other mathematical operations.

## 3. What is the purpose of using a measure of a set?

The purpose of using a measure of a set is to quantify the size or extent of a set in a meaningful way. This can allow for comparisons between sets, the ability to make predictions or inferences, and the development of mathematical models and theories.

## 4. How is a measure of a set related to probability?

A measure of a set can be used to determine the probability of an event occurring. In probability theory, a measure of a set is known as a probability measure and is used to assign probabilities to certain outcomes or events within a given set.

## 5. What are some common measures of a set?

Some common measures of a set include length, area, volume, and probability. Other measures may include measures of uncertainty, distance, or mass. The specific measure used will depend on the context and application of the set being studied.

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