. 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.
In this section the extremal length is calculated in several examples. The first three of these examples are actually useful in applications of extremal length.
Extremal distance in rectangle
Fix some positive numbers , and let be the rectangle. Let be the set of all finitelength curves that cross the rectangle left to right,in the sense that is on the left edge of the rectangle, and is on the right edge .(The limits necessarily exist, because we are assuming that has finite length.) We will now prove that in this case:
First, we may take on . This gives and . The definitionof as a supremum then gives .
The opposite inequality is not quite so easy. Consider an arbitraryBorel-measurable such that.For , let (where we are identifying with the complex plane).Then , and hence .The latter inequality may be written as:Integrating this inequality over implies:.Now a change of variable and an application of the Cauchy-Schwarz inequality give:. This gives . Therefore, , as required.
As the proof shows, the extremal length of is the same as the extremallength of the much smaller collection of curves .
It should be pointed out that the extremal length of the family of curves that connect the bottom edge of to the top edge of satisfies, by the same argument. Therefore, .It is natural to refer to this as a duality property of extremal length, and a similar duality propertyoccurs in the context of the next subsection. Observe that obtaining a lower bound on is generally easier than obtaining an upper bound, since the lower bound involveschoosing a reasonably good and estimating ,while the upper bound involves proving a statement about all possible . For this reason,duality is often useful when it can be established: when we know that ,a lower bound on translates to an upper bound on .
Extremal distance in annulus
Let and be two radii satisfying