Definition of Extremal Length
To define extremal length, we need to first introduce several related quantities. Let be an open set in the complex plane. Suppose that is a collection of rectifiable curves in . If is Borel-measurable, then for any rectifiable curve we let
denote the -length of , where denotes the Euclidean element of length. (It is possible that .) What does this really mean? If is parameterized in some interval, then is the integral of the Borel-measurable function with respect to the Borel measure on for which the measure of every subinterval is the length of the restriction of to . In other words, it is the Lebesgue-Stieltjes integral, where is the length of the restriction of to . Also set
The area of is defined as
and the extremal length of is
where the supremum is over all Borel-measureable with . If contains some non-rectifiable curves and denotes the set of rectifiable curves in, then is defined to be .
The term modulus of refers to .
The extremal distance in between two sets in is the extremal length of the collection of curves in with one endpoint in one set and the other endpoint in the other set.
Read more about this topic: Extremal Length
Famous quotes containing the words definition of, definition and/or length:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.”
—Henry David Thoreau (1817–1862)