Wait, there are a lot of ways in computer science to choose points randomly, and points randomly with constraints (here belong to unit sphere) and that could change the result, no ?
>We’ll be ‘randomly choosing’ lots of points on spheres of various dimensions. Whenever we do this, I mean that they’re chosen independently, and uniformly with respect to the unique rotation-invariant Borel measure that’s a probability measure on the sphere. In other words: nothing sneaky, just the most obvious symmetrical thing!
There are an infinite number of probability distributions over most objects, yes, but there's also often a good default that is the "uniform" distribution. That's what they're talking about here.
As an aside, there's an interesting way to generate a uniform distribution on a sphere. It uses the fact that the joint distribution of several independent standard Gaussians has rotational symmetry. So to generate a random point on the surface of the n-sphere, you can sample n+1 numbers independently from the standard Gaussian and then normalise them so that their squares sum to 1 (if they're all 0 then you have to reroll).
In fact the result has little to do with computer science - it is purely a mathematical result. The way points are sampled (uniform distribution) is clarified in the text
