On uniformly randomly choosing a pure quantum state If I have a Hilbert space $\mathcal{H}_A$, how can one uniformly randomly choose a pure quantum state in this space? 
I believe the answer is to take the state $\vert 0\rangle$ and apply a random unitary to it. This random unitary is chosen according to a measure called the Haar measure. I do not have a measure theory background so I'm struggling to understand this and the mathematical discussions about the Haar measure are hard to follow. 
Is there a physicist's guide to understanding how to pick a random quantum state from some Hilbert space such that every state is equiprobable?
 A: I don't have a complete answer, but this may help. 
A Hilbert space is a complete normed vector space. The norm gives vectors a length and an angle between pairs of vectors. From this, you can get volumes and areas. You want to choose states whose norm is $1$. That is, you want to pick vectors uniformly from the unit sphere. Dividing the sphere into small patches of area $\epsilon$, you want the same probability of being in any patch. 
Googling "choosing points uniformly on n-dimensional spheres" turns up a number of solutions for finite dimensional Hilbert spaces. Wolfram has this one. Here is something on Geometry of high-dimensional space
But if you have an infinite dimensional Hilbert space, it gets trickier. You need a Haar measure. According to Wikipedia, 

The Haar measure assigns an "invariant volume" to subsets of locally
  compact topological groups, consequently defining an integral for
  functions on those groups.

This doesn't sound very helpful. But a Hilbert space is a locally compact topological group. It means a Haar measure allows you to define integration over subsets of an infinite dimensional Hilbert space.
But I don't know if it is possible possible to choose a uniform distribution over an infinite dimensional hypersphere. From the Wikipedia article on the n-sphere, you can see the surface area of the unit sphere goes to 0 as the dimension increases. If all patches on the sphere have area $0$, it is hard to choose patches of comparable size. 
Part of the trickiness is that you can form the set of all ordered infinity-tuples $(x_0, x_1, ...)$. But not all of these represent points in an infinite dimensional Hilbert space. A Hilbert space has a norm, which means that all vectors have a finite length. The usual norm is $\sqrt{{x_0}^2 + {x_1}^2 + ...}$ This means  $(1/2, 1/4, ...)$ is a vector in the space, but $(1, 1, ...)$ is not.  
Suppose you pick some patch around $(1,0,0,...)$, and another very similar patch around $(0,1,0,...)$, and so on. You now have an infinite number of patches that all must have equal probability. The probability of each is infinitesimal.
