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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.

작업 예제 · integrate 1/10 dx from 2 to 5

Calculus · Definite-Integrals

Integrate 1/10 from 2 to 5

\int_{2}^{5} \frac{1}{10}\, dx

단계별로

  1. \int_{2}^{5} \frac{1}{10}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int \frac{1}{10}\, dx = \frac{x}{10}

    The integral of a constant c is c·x.

  3. F(5) - F(2) = \left(\frac{1}{2}\right) - \left(\frac{1}{5}\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{3}{10}

    Simplify.

답을 드러내세요
\frac{3}{10}

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어떻게: σ-algebras and measures

  1. First find an antiderivative F, then evaluate F(b) − F(a).
  2. The integral of a constant c is c·x.
  3. Fundamental theorem of calculus: plug in the limits.
  4. Simplify.

사람들이 묻는 질문

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.

에 더 Measure Theory

The Lebesgue integral and convergence theorems