Abstract: A conformal restriction measure is a probability measure which is used to describe the law of a random connected subset in a simply connected domain that satisfies a certain conformal restriction property. Usually there are three kinds of conformal restriction measures:one (called the chordal restriction measure) has two given boundary points of the random set, the second (called the radial restriction measure) has one boundary point and one interior point in the random set, and the third (called the tri-chordal restriction measure) has three boundary points in the random set. In this article, we will define a new probability measure such that the random set associated to it contains one given interior point and does not intersect with the boundary. Furthermore, we will show that this measure can be characterized by one parameter; we will also construct this one-parameter family of measures in two ways and obtain several properties.

Key words: self-avoiding loops, conformal restriction measure, SLE, Brownian loop measure