Understanding Lebesgue Measure: Example of Open Intervals on [0,1]

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In summary, there is confusion about lebesgue measure and its relation to the rational numbers. It is known that the lebesgue outer measure of the rationals is 0. This implies that there must be a set of open intervals on [0, 1] that contains the rationals and has a total length less than 1, according to the definition of lebesgue outer measure. However, it is unclear how this could be possible. After further consideration, the confusion was resolved.
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tomprice
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I am confused about lebesgue measure.

I have heard that the lebesgue outer measure of the rational numbers is 0.

So could someone please give an example of a set of open intervals such that:

a. The union of these intervals contains the rational numbers on [0, 1]

b. The sum of lengths of these intervals is less than 1.

The definition of lebesgue outer measure implies that, if the lebesgue outer measure of the rationals is 0, then such a set of open intervals must exist. But I am flabberghasted as to how this could be so.

Thanks very much in advance.
 
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false alarm

Nevermind I got it.
 

FAQ: Understanding Lebesgue Measure: Example of Open Intervals on [0,1]

What is Lebesgue Measure?

Lebesgue Measure is a mathematical concept used in measure theory to assign a numerical value to subsets of a given set. It is named after French mathematician Henri Lebesgue and is used to measure the length, area, or volume of a set.

How is Lebesgue Measure different from other measures?

Lebesgue Measure is different from other measures, such as the Riemann integral, because it takes into account more complex sets, such as uncountable sets and sets with non-measurable points. It also allows for a more precise measurement of sets, as it considers the distribution and density of points within a set.

Why is the example of open intervals used to explain Lebesgue Measure?

The example of open intervals on [0,1] is commonly used to explain Lebesgue Measure because it is a simple and intuitive way to understand the concept. It also illustrates how Lebesgue Measure differs from other measures, as open intervals are not measurable using the Riemann integral but can be measured using Lebesgue Measure.

How is Lebesgue Measure used in other fields of science?

Lebesgue Measure has many applications in various fields of science, such as physics, economics, and probability theory. It is used to measure the size and shape of objects, to calculate probabilities and expected values, and to analyze the distribution of data.

Can Lebesgue Measure be extended to higher dimensions?

Yes, Lebesgue Measure can be extended to higher dimensions, such as 3-dimensional space. In higher dimensions, it is called Lebesgue-Borel Measure and is used to measure the volume of objects and sets in higher-dimensional spaces.

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