Concerning the difference between our definitions of measurement, I was referring to the prior probability of making a measurement given the true state, which is P(z|x), or, if you prefer, f(z,x) = P(Z=z|X=x). This PDF is Normal(Hx, R). The distribution you refer to is that of the image of the true state in measurement space (Hx), which is Normal(z, R).

Concerning the general question of dimensionality and numbers vs. functions, I'm afraid I still don't understand the fundamental issue. I agree that probability notation isn't the best, but each of those terms can be replaced with a Gaussian (or in the case of the denominator, an integral of the product of two Gaussians). You end up with a function in terms of x_k and the sequence of z's. You can then choose which variables are free or bound and evaluate the function as your application requires. If all variables are bound (the case that you know the measurement z_k, and are evaluating the posterior probability of the state at a given point), all the terms are numbers (yes, densities), and there are no dimensional issues. If you leave x_k free, you end up with two functions in the numerator, and a number in the denominator. Those two functions are Mahalanobis distances (which are scalars) in an exponent, and again, no dimensional issues. In general, such operations with PDFs represent various combinations and manipulations of event spaces that need not be commensurate. Computing the PDF of a coin flip given the distribution of all of the positions and velocities of the molecules of air through which the coin flies requires comparing a two-state PMF to a virtually infinite dimensional PDF. Notationally speaking, no problem.