Editing Borel–Kolmogorov paradox (section)

== Explanation and implications ==
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 limit of events ''R''<sub>''ab''</sub> = {''φ'' : ''a'' < ''φ'' < ''b''} which are horizontal rings (curved surface zones of [[spherical segment]]s) 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 a limit of the events ''L''<sub>''cd''</sub> = {''λ'' : ''c'' < ''λ'' < ''d''}, which are [[Spherical lune|lunes]] (vertical wedges), consisting of all points whose longitude varies between ''c'' and ''d''.  So although ''P''(''λ'' ∈ ''E'' | ''φ'' = 0) and ''P''(''φ'' ∈ ''F'' | ''λ'' = 0) each provide a probability distribution on a great circle, one of them is defined using limits of rings, and the other using limits of lunes.  Since rings and lunes have different shapes, it should be less surprising that ''P''(''λ'' ∈ ''E'' | ''φ'' = 0) and ''P''(''φ'' ∈ ''F'' | ''λ'' = 0) have different distributions.

{{Quote
 | The concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible. For we can obtain a probability distribution for [the latitude] on the meridian circle only if we regard this circle as an element of the decomposition of the entire spherical surface onto meridian circles with the given poles
| [[Andrey Kolmogorov]]<ref>Originally [[#kol1933|Kolmogorov (1933)]], translated in [[#kol1956|Kolmogorov (1956)]]. Sourced from [[#pol2002|Pollard (2002)]]</ref>
}}
{{Quote
 | … the term 'great circle' is ambiguous until we specify what limiting operation is to produce it. The intuitive symmetry argument presupposes the equatorial limit; yet one eating slices of an orange might presuppose the other.
 | [[E.T. Jaynes]]<ref name=Jaynes/>
}}