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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.
Çözümlü örnek · integrate 1/10 dx from 2 to 5
Adım adım
- \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.
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: 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.
Sık sorulan sorular
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.