Enveloping von Neumann algebra

In mathematics, contour sets generalize and formalize the everyday notions of

• everything superior to something
• everything superior or equivalent to something
• everything inferior to something
• everything inferior or equivalent to something.

Formal definitions

Given a relation on pairs of elements of set $X$

$\succcurlyeq ~\subseteq ~X^{2}$

and an element $x$ of $X$

$x\in X$

The upper contour set of $x$ is the set of all $y$ that are related to $x$:

$\left\{y~\backepsilon ~y\succcurlyeq x\right\}$

The lower contour set of $x$ is the set of all $y$ such that $x$ is related to them:

$\left\{y~\backepsilon ~x\succcurlyeq y\right\}$

The strict upper contour set of $x$ is the set of all $y$ that are related to $x$ without $x$ being in this way related to any of them:

$\left\{y~\backepsilon ~(y\succcurlyeq x)\land \lnot (x\succcurlyeq y)\right\}$

The strict lower contour set of $x$ is the set of all $y$ such that $x$ is related to them without any of them being in this way related to $x$:

$\left\{y~\backepsilon ~(x\succcurlyeq y)\land \lnot (y\succcurlyeq x)\right\}$

The formal expressions of the last two may be simplified if we have defined

$\succ ~=~\left\{\left(a,b\right)~\backepsilon ~\left(a\succcurlyeq b\right)\land \lnot (b\succcurlyeq a)\right\}$

so that $a$ is related to $b$ but $b$ is not related to $a$, in which case the strict upper contour set of $x$ is

$\left\{y~\backepsilon ~y\succ x\right\}$

and the strict lower contour set of $x$ is

$\left\{y~\backepsilon ~x\succ y\right\}$

Contour sets of a function

In the case of a function $f()$ considered in terms of relation $\triangleright$, reference to the contour sets of the function is implicitly to the contour sets of the implied relation

$(a\succcurlyeq b)~\Leftarrow ~[f(a)\triangleright f(b)]$

Examples

Arithmetic

Consider a real number $x$, and the relation $\geq$. Then

• the upper contour set of $x$ would be the set of numbers that were greater than or equal to $x$,
• the strict upper contour set of $x$ would be the set of numbers that were greater than $x$,
• the lower contour set of $x$ would be the set of numbers that were less than or equal to $x$, and
• the strict lower contour set of $x$ would be the set of numbers that were less than $x$.

Consider, more generally, the relation

$(a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]$

Then

• the upper contour set of $x$ would be the set of all $y$ such that $f(y)\geq f(x)$,
• the strict upper contour set of $x$ would be the set of all $y$ such that $f(y)>f(x)$,
• the lower contour set of $x$ would be the set of all $y$ such that $f(x)\geq f(y)$, and
• the strict lower contour set of $x$ would be the set of all $y$ such that $f(x)>f(y)$.

It would be technically possible to define contour sets in terms of the relation

$(a\succcurlyeq b)~\Leftarrow ~[f(a)\leq f(b)]$

though such definitions would tend to confound ready understanding.

In the case of a real-valued function $f()$ (whose arguments might or might not be themselves real numbers), reference to the contour sets of the function is implicitly to the contour sets of the relation

$(a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]$

Note that the arguments to $f()$ might be vectors, and that the notation used might instead be

$[(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]$

Economic

In economics, the set $X$ could be interpreted as a set of goods and services or of possible outcomes, the relation $\succ$ as strict preference, and the relationship $\succcurlyeq$ as weak preference. Then

• the upper contour set, or better set, of $x$ would be the set of all goods, services, or outcomes that were at least as desired as $x$,
• the strict upper contour set of $x$ would be the set of all goods, services, or outcomes that were more desired than $x$,
• the lower contour set, or worse set, of $x$ would be the set of all goods, services, or outcomes that were no more desired than $x$, and
• the strict lower contour set of $x$ would be the set of all goods, services, or outcomes that were less desired than $x$.

Such preferences might be captured by a utility function $u()$, in which case

• the upper contour set of $x$ would be the set of all $y$ such that $u(y)\geq u(x)$,
• the strict upper contour set of $x$ would be the set of all $y$ such that $u(y)>u(x)$,
• the lower contour set of $x$ would be the set of all $y$ such that $u(x)\geq u(y)$, and
• the strict lower contour set of $x$ would be the set of all $y$ such that $u(x)>u(y)$.

Complementarity

On the assumption that $\succcurlyeq$ is a total ordering of $X$, the complement of the upper contour set is the strict lower contour set.

$X^{2}\backslash \left\{y~\backepsilon ~y\succcurlyeq x\right\}=\left\{y~\backepsilon ~x\succ y\right\}$

$X^{2}\backslash \left\{y~\backepsilon ~x\succ y\right\}=\left\{y~\backepsilon ~y\succcurlyeq x\right\}$

and the complement of the strict upper contour set is the lower contour set.

$X^{2}\backslash \left\{y~\backepsilon ~y\succ x\right\}=\left\{y~\backepsilon ~x\succcurlyeq y\right\}$

$X^{2}\backslash \left\{y~\backepsilon ~x\succcurlyeq y\right\}=\left\{y~\backepsilon ~y\succ x\right\}$

See also

• Epigraph
• Hypograph

Bibliography

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (Template:LCC), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)