Translating between momentum eigenstate and position eigenstate Say a particle is in an eigenstate of momentum:
$$\phi_{\mathbf p} = N  e^{i \mathbf p x}$$ 
Then, according to my lecture notes, apparently a position measurement renders
$$\psi = \delta ^{(3)} (\mathbf x - \mathbf{x_0})$$
While this makes sense intuitively (in terms of what I would expect without doing any math) I don't see how this translation was made. Does it have to do with a fourier transform pair between a plane wave and dirac delta function? How does one go from one to the other?
Snippet of my lecture notes are below.

 A: It doesn't have to do with anything fancy. (11) just comes from the fact that measurement done on the system must entail a collapse of that system into one of the observable's eigenstates. (10) doesn't really matter in this context: Any other wavefunction would collapse in the same way. I'm guessing they specifically chose a momentum eigenstate to make a point about commuting observables.
A: $\def\bx{\mathbf x} \def\hA{\hat A}$
Your difficulty arises, IMHO, from a questionable approach many
teachers take when introducing QM. Basically it can be recognized in
the attempt to give a "physical" meaning to such things as
commutators, uncertainty and so on. And in not making a clear
distinction between two levels of the theory:


*

*Mathematical background (Hilbert space, operators, eigenvalues ...)

*Their physical interpretation, given by totally separate axioms.
The main point is just the one you are stressing: what does a measurement
mean? The answer should not be given within the mathematical
development of theory. According to presently held interpretation of
QM the act of measurement has no representative in the mathematical
structure of QM. So there is no "Fourier transform" as you would like.
There is a postulate:
Given a state - represented e.g. by a normalized wavefunction
$\psi(\bx)$ - the measurement of an observable $A$ - represented by a
self-adjoint operator $\hA$ on the Hilbert space of wavefunctions - is
an eigenvalue $a$ of $\hA$. Immediately after measurement the system's
state is the eigenfunction $\phi_a(\bx)$ belonging to that eigenvalue.
Which eigenvalue and eigenfunction will result is undetermined, only
its probability  being known
$$\left|\int\!d^{(3)}\bx\;\phi_a^*(\bx)\,\psi(\bx)\right|^2\!.$$
Caution. The above statement is a simplification, as it doesn't
take into account the case of degenerate eigenvalues.
The most obvious mark ot that approach you'll find about uncertainty
principle. You'll see UP stated as a limitation of simultaneous
measurements, even to be justifed by analyzing paradigmatic
experiments (like the famous Heisenberg microscope). Actually it is
instead nothing but a theorem of the mathematical theory. Given the
above postulate, UP follows.
It's true that the interpretation of UP as ensuing from intrinsic experimental
limitations of quantum objects was given by Heisenberg himself, but it was almost a century ago. We are not obliged, when teaching QM in present times, to retrace the hard way their creators had to follow to understand it.
