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− | A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254901.png" /> has (Lebesgue) content zero if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254902.png" /> there is a finite set of closed rectangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254903.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254906.png" /> is [[Lebesgue measure|Lebesgue measure]]. | + | 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 $\sum_i \mu(U_i)$, where $\mu$ is [[Lebesgue measure]]. |
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− | More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254907.png" /> be a space equipped with a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254908.png" /> of subsets such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c0254909.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549010.png" /> need not be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549011.png" />-ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549012.png" /> need not be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549013.png" />). Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549015.png" /> be given such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549018.png" /> for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549019.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549020.png" /> is additive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549021.png" />. Such a function is called a content, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549022.png" /> is the content of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549023.png" />. | + | 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 $\mathcal{E}$). 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 additive on $\mathcal{E}$. Such a function is called a content, and $\gamma(A)$ is the content of $A$. |
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− | Define a rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549024.png" /> as a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549025.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549026.png" /> are bounded closed, open or half-closed intervals, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549028.png" /> is the length of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549029.png" />. Define an elementary set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549030.png" /> to be a finite union of rectangles. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549031.png" /> be the collection of all elementary sets. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549032.png" /> can be written as a finite disjoint union of rectangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549033.png" />; then define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549034.png" />. This defines a content on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549035.png" /> called Jordan content. | + | 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. |
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− | Given a content <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549037.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549039.png" />, one defines | + | 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|outer measure]] on $S$. |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549040.png" /></td> </tr></table> | + | ====References==== |
| + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.F. Randolph, "Basic real and abstract analysis" , Acad. Press (1968)</TD></TR> |
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> M.M. Rao, "Measure theory and integration" , Interscience (1987)</TD></TR> |
| + | </table> |
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− | where the infimum is taken over all finite sums such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549042.png" />; also one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549043.png" />. This defines an [[Outer measure|outer measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025490/c02549044.png" />.
| + | {{TEX|done}} |
− | | |
− | ====References====
| |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.F. Randolph, "Basic real and abstract analysis" , Acad. Press (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.M. Rao, "Measure theory and integration" , Interscience (1987)</TD></TR></table>
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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 $\sum_i \mu(U_i)$, 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 $\mathcal{E}$). 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 additive 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) |