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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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

Àwọn ìṣàmúlò-ètò

\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.

Fi àwọn àgbèwọlé hàn

\frac{3}{10}

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Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Measure Theory

The Lebesgue integral and convergence theorems