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σ-algebras and measures
Which sets can be measured, and the countable additivity that makes limits work.
A σ-algebra is a family of sets closed under complements and countable unions; a measure assigns sizes additively over countable disjoint unions. Lebesgue measure gives intervals their length and extends to far stranger sets. Picture it: covering a set by intervals and taking the cheapest total length. Think it: not every set is measurable (Vitali), which is why the σ-algebra is part of the definition.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · integrate 1/10 dx from 2 to 5
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
- \int_{2}^{5} \frac{1}{10}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
- \int \frac{1}{10}\, dx = \frac{x}{10}
The integral of a constant c is c·x.
- F(5) - F(2) = \left(\frac{1}{2}\right) - \left(\frac{1}{5}\right)
Fundamental theorem of calculus: plug in the limits.
- = \frac{3}{10}
Simplify.
ເປີດເຜີຍຄຳຕອບ
ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
ວິທີການ: σ-algebras and measures
- First find an antiderivative F, then evaluate F(b) − F(a).
- The integral of a constant c is c·x.
- Fundamental theorem of calculus: plug in the limits.
- Simplify.
ຄໍາຖາມທີ່ຄົນຖາມ
What is wrong with the Riemann integral?
It fails on functions that oscillate too much, and it does not interact well with limits: the limit of integrable functions need not be integrable. Lebesgue's integral fixes both.