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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.

Exemplo trabalhado: 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

Passo a passo

  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.

Revelar a resposta
\frac{3}{10}

Symbols used here

\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.
\mu(A),\ \sigma\text{-algebra}
measure of A
Size of a set; the family of sets that can be measured.
\|f\|_p,\ \langle f, g \rangle
p-norm, inner product
Length of a function; the generalised dot product.

How to: 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.

Questions people ask

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.

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