Is it enough to show that it is a Lebesgue measure?

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In summary, a Lebesgue measure is a mathematical concept used to measure the size or volume of subsets of n-dimensional Euclidean space, named after mathematician Henri Lebesgue. It differs from other measures by considering the entire set of real numbers and allows for the measurement of complex sets like fractals. Showing that something has a Lebesgue measure is important for using powerful mathematical tools and measuring non-traditional sets. To determine if something has a Lebesgue measure, it must be a measurable set according to Carathéodory's criterion. Not all sets have a Lebesgue measure, as some may be non-measurable or have a measure of zero.
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mathmari
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Hey! :eek:

I want to show that when $\mu$ is a Borel measure in $\mathbb{R}$ with $\mu([0,1))=1$, which is a translation invariant, then it is also a Lebesgue measure.

I have shown that $\mu([a,b))=b-a, \forall a,b \in \mathbb{Q}$.

Is it enough to show that $\mu$ is a Lebesgue measure?? (Wondering)
 
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If two Borel measures agree on intervals of the form $[a,b)$ then they agree on anything in the $\sigma$-algebra which is generated by those intervals.
 

FAQ: Is it enough to show that it is a Lebesgue measure?

What is a Lebesgue measure?

A Lebesgue measure is a mathematical concept used to measure the size or volume of subsets of n-dimensional Euclidean space. It is named after the mathematician Henri Lebesgue.

How is a Lebesgue measure different from other measures?

A Lebesgue measure differs from other measures, such as the Riemann measure, in that it takes into account the entire set of real numbers and not just the endpoints of intervals. It also allows for the measurement of more complex sets, such as fractals.

Why is it important to show that something has a Lebesgue measure?

Showing that something has a Lebesgue measure is important because it allows for the use of powerful mathematical tools, such as the Lebesgue integral, to analyze and make calculations about the set. It also provides a way to measure sets that may not have a traditional volume or size.

How is it determined that something has a Lebesgue measure?

To determine if something has a Lebesgue measure, one must show that it is a measurable set. This means it can be approximated by a sequence of simple sets whose measures converge to the measure of the original set. This is known as the Carathéodory's criterion.

Can any set have a Lebesgue measure?

No, not all sets have a Lebesgue measure. For example, non-measurable sets, such as Vitali sets, cannot have a Lebesgue measure. Additionally, some sets may have a Lebesgue measure of zero, meaning they have no size or volume according to the Lebesgue measure.

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