Abstract
There is a motivation to generalize the notion of length to another concept that could capture length in a more general setting. In principle, the measure of a set \(A\subseteq \mathbb {R}\) should refer to the size of A, and it should agree with the natural properties of length if the settings were reduced to length.
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- 1.
In fact, this is what motivated Lebesgue in the first place to define the inner measure.
- 2.
We have to admit that accepting AC will make our mathematical life easier and proofs and techniques become more efficient, because it has numerous applications. Still, on the other hand, mathematics will not be miserable without AC. Paradoxes and disasters will come out whether we accept AC or deny it. Whether we should accept or deny the AC is entirely beyond this book’s scope, and the interested reader may consult books on Set Theory and Mathematical Philosophy.
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Khanfer, A. (2023). Measure Theory. In: Measure Theory and Integration. Springer, Singapore. https://doi.org/10.1007/978-981-99-2882-8_1
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DOI: https://doi.org/10.1007/978-981-99-2882-8_1
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