Namespaces
Variants
Actions

Content

From Encyclopedia of Mathematics
Jump to: navigation, search

A set $A \subset \mathbb{R}^n$ has (Lebesgue) content zero if for all $\epsilon > 0$ there is a finite set of closed rectangles $U_1,\ldots,U_n$ such that $A \subset \bigcup_i U_i$ and $\sum_i \mu(U_i) < \epsilon$, where $\mu$ is Lebesgue measure.

More generally, let $S$ be a space equipped with a ring $\mathcal{E}$ of subsets such that $S \subset \bigcup_{A \in \mathcal{E}} A$ ($\mathcal{E}$ need not be a $\sigma$-ring and $S$ need not be in $\mathcal{E}$: cf. Ring of sets). Let a function $\gamma$ on $\mathcal{E}$ be given such that $0 \le \gamma(A) < \infty$ for all $A \in \mathcal{E}$, $\gamma(A) > 0$ for at least one $A \in \mathcal{E}$ and such that $\gamma$ is an additive function on $\mathcal{E}$. Such a function is called a content, and $\gamma(A)$ is the content of $A$.

Define a rectangle $R \subset \mathbb{R}^n$ as a product $I_1 \times \cdots \times I_n$, where the $I_i$ are bounded closed, open or half-closed intervals, and let $|R| = \prod_i l(I_i)$, where $l(I_i)$ is the length of the interval $I_i$. Define an elementary set in $\mathbb{R}^n$ to be a finite union of rectangles. Let $\mathcal{E}$ be the collection of all elementary sets. Each $A \in \mathcal{E}$ can be written as a finite disjoint union of rectangles $ = \bigcup_j R_j$; then define $\gamma(A) = \sum_j |R_j|$. This defines a content on $\mathcal{E}$ called Jordan content.

Given a content $\gamma$ on $\mathcal{E}$ and any $A \subset S$, $A \neq \emptyset$, one defines $$ \mu^*(A) = \inf \sum_n \gamma(A_n) $$ where the infimum is taken over all finite sums such that $A \subset \bigcup A_n$, $A_n \in \mathcal{E}$; also one sets $\mu^*(\emptyset) = 0$. This defines an outer measure on $S$.

References

[a1] J.F. Randolph, "Basic real and abstract analysis" , Acad. Press (1968)
[a2] M.M. Rao, "Measure theory and integration" , Interscience (1987)

Comment

For the content of a polynomial, see Primitive polynomial.

How to Cite This Entry:
Content. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Content&oldid=42037