---
tag: 9RMS0ZJ618
kind: counterexample
name: Borel–Kolmogorov paradox
msc: [60A05, 60A10]
uses: [059TJN0NZ1, 01QFQAQ2PA, 0ZY9DX8FBF, 0ZN9EZTEM6, PHHT5W3BEJ]
sources:
  - url: https://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox
same_as:
  wikidata: Q3363312
  wikipedia: Borel–Kolmogorov paradox
---

## Counterexample

Let $P$ be the uniform [[probability measure|059TJN0NZ1]] on the unit sphere $S^2 \subset \R^3$.
Describe a point by its longitude $\lambda \in (-\pi, \pi]$ and its latitude $\varphi \in [-\pi/2, \pi/2]$, as in [[spherical coordinates|01QFQAQ2PA]].
The area element of $S^2$ is $\cos \varphi \, d\lambda \, d\varphi$, so under $P$ the pair $(\lambda, \varphi)$ has the [[density|0ZY9DX8FBF]] $\frac{1}{4\pi} \cos \varphi$.
Hence $\lambda$ and $\varphi$ are independent, $\lambda$ is uniform on $(-\pi, \pi]$, and $\varphi$ has density $\frac{1}{2} \cos \varphi$.

Condition on the event $\{\varphi = 0\}$, the equator.
The [[conditional distribution|0ZN9EZTEM6]] of $\lambda$ given $\varphi = 0$ is the law of $\lambda$, which is uniform.
So the point is uniform on the equator.

Now let $\Theta = \lambda \bmod \pi \in [0, \pi)$ and condition on the event $\{\Theta = 0\}$, the great circle through the two poles.
Since $\Theta$ is independent of $\varphi$, the conditional distribution of $\varphi$ given $\Theta = 0$ still has density $\frac{1}{2} \cos \varphi$.
So the point is not uniform on this great circle: it is rarely near the poles.

A rotation of the sphere maps the equator onto the great circle through the poles and preserves $P$.
Yet the two conditional laws on these two great circles are not images of each other under the rotation.

## Remarks

Both events have probability $0$.
The conditional law given a null event depends on the random variable whose level set it is, not on the event alone.
A [[regular conditional probability|PHHT5W3BEJ]] given $\varphi$, or given $\Theta$, is defined only up to null sets of values, so it fixes no answer for a single null event.

## Explanations

### L1 human

Pick a point at random on the surface of the Earth, and suppose you are told that it lies on a given great circle.
One may expect the point to be uniform on that circle.
If the circle is seen as a line of constant latitude (the equator), the answer is uniform.
If the same kind of circle is seen as a line of constant longitude (a meridian circle), points near the poles are less likely, because meridians crowd together there.
The lesson is that one cannot condition on an event of probability zero by itself.
One must say which family of events it belongs to, and this is what a [[regular conditional probability|PHHT5W3BEJ]] does.