Borel–Kolmogorov paradox

In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect to an event of probability zero (also known as a null set). It is named after Émile Borel and Andrey Kolmogorov. Suppose that a random variable has a uniform distribution on a unit sphere. What is its conditional distribution on a great circle? Because of the symmetry of the sphere, one might expect that the distribution is uniform and independent of the choice of coordinates. However, two analyses give contradictory results. First, note that choosing a point uniformly on the sphere is equivalent to choosing the longitude uniformly from and choosing the latitude from with density . Then we can look at two different great circles: If the coordinates are chosen so that the great circle is an equator (latitude ), the conditional density for a longitude defined on the interval is If the great circle is a line of longitude with , the conditional density for on the interval is One distribution is uniform on the circle, the other is not. Yet both seem to be referring to the same great circle in different coordinate systems. Many quite futile arguments have raged — between otherwise competent probabilists — over which of these results is 'correct'. In case (1) above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P(λ ∈ E | φ = 0). Elementary probability theory suggests this can be computed as P(λ ∈ E and φ = 0)/P(φ = 0), but that expression is not well-defined since P(φ = 0) = 0. Measure theory provides a way to define a conditional probability, using the family of events Rab = {φ : a < φ < b} which are horizontal rings consisting of all points with latitude between a and b. The resolution of the paradox is to notice that in case (2), P(φ ∈ F | λ = 0) is defined using the events Lab = {λ : a < λ < b}, which are lunes (vertical wedges), consisting of all points whose longitude varies between a and b.