Content

A set $A\subset {\mathbb{R}}^n$ has (Lebesgue) content zero if for all $ϵ>0$ there is a finite set of closed rectangles $U_1,\dots ,U_n$ such that $A\subset \bigcup _i{U}_i$ and $\sum _i\mu (U_i)<ϵ$, 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 $0\le \gamma (A)<\mathrm{\infty }$ 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 $A\subset S$, $A\ne \mathrm{\varnothing }$, one defines $${\mu }^{\ast }(A)=inf\sum _n\gamma (A_n)$$ where the infimum is taken over all finite sums such that $A\subset \bigcup A_n$, $A_n\in \mathcal{E}$; also one sets ${\mu }^{\ast }(\mathrm{\varnothing })=0$. This defines an outer measure on $S$.

References[edit]

Comment[edit]

For the content of a polynomial, see Primitive polynomial.