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Lebesgue integral vs Riemann integral

| 🗖 nosplit | ↑ parent "Lebesgue integral" | words: 201 | descendant words: 272 | descendants: 1
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Advantages over Riemann:
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Video 174. "Riemann integral vs. Lebesgue integral" by The Bright Side Of Mathematics (2018) Source.
https://youtube.com/watch?v=PGPZ0P1PJfw&t=808 shows how Lebesgue can be visualized as a partition of the function range instead of domain, and then you just have to be able to measure the size of pre-images.
One advantage of that is that the range is always one dimensional.
But the main advantage is that having infinitely many discontinuities does not matter.
Infinitely many discontinuities can make the Riemann partitioning diverge.
But in Lebesgue, you are instead measuring the size of preimage, and to fit infinitely many discontinuities in a finite domain, the size of this preimage is going to be zero.
So then the question becomes more of "how to define the measure of a subset of the domain".
Which is why we then fall into measure theory!
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Ancestors

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