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.
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