Is Uniqueness Necessary in Mathematical Measures?

In summary, the book "Measures, integrals and Martingales" by Schilling discusses the uniqueness and existence of the Lebesgue measure in mathematics. While there are other measures with similar properties, the Lebesgue measure is the only one that has the specific property of the volume of a box in R^n being the product of the length of its sides. It is considered important because of this uniqueness, but it is not clear why this is necessarily a good thing. The existence of the measure implies that it is well defined on all sets, given the properties that define it.
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aaaa202
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My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure with the same properties as the Lebesgue measure? What is necessarily so good about the uniqueness property?
Also it has a theorem of its existence. But what does existence of a measure imply? That it is well defined on all sets given the properties that defines it?
 
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  • #2
aaaa202 said:
My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure with the same properties as the Lebesgue measure? What is necessarily so good about the uniqueness property?
Also it has a theorem of its existence. But what does existence of a measure imply? That it is well defined on all sets given the properties that defines it?

There are lots of other, non-Lebesque measures with the same basic properties, so you need to tell us exactly how the Lebesgue measure differs from these others. We don't know what your book says (and we don't even know the title/author of the book), so you need to fill in the details for us.
 
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It is "Measures, integrals and Martingales" by Schilling. There exists only one measure, the Lebesgue measure, with the property that the volume of a box in R^n is the product of the length of its sides, which in turn are b-a, b the upper point and a the lower. I just wanted to know why it is so important that it is unique.
 

What is uniqueness in scientific research?

Uniqueness in scientific research refers to the originality and distinctiveness of a particular study or experiment. It means that the research is not a replication of previous work and offers new insights or findings.

Why is uniqueness important in scientific research?

Uniqueness is important in scientific research because it contributes to the advancement of knowledge and understanding in a particular field. It also ensures that the research is not biased or influenced by previous studies, allowing for more reliable and accurate results.

What is meant by existence in scientific research?

Existence in scientific research refers to the presence or occurrence of a particular phenomenon or entity. It is the confirmation that something exists and can be observed or measured through scientific methods.

How is existence determined in scientific research?

Existence is determined in scientific research through empirical evidence and rigorous testing. Scientists use various methods such as experiments, observations, and data analysis to confirm the existence of a particular concept, theory, or phenomenon.

What is the relationship between uniqueness and existence in scientific research?

The relationship between uniqueness and existence in scientific research is that both are essential for producing valid and reliable results. Uniqueness ensures that the research is original and not influenced by previous studies, while existence confirms that the phenomenon being studied is real and can be observed or measured.

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