Q:

Observing a system in an energy eigenstate when the eigenstate is not normalized

In the following notes from an MIT OCW course, Zweibach claims that energy eigenstates are not necessarily normalized.
https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_01.pdf

A solution $\psi(x)$ associated with an energy $E$ is called an energy eigenstate of energy $E$. [...] We do not impose the requirement that $ψ(x)$ be normalizable. This would be too restrictive. There are energy eigenstates that are not normalizable. Momentum eigenstates of a free particle are also not normalizable. Solutions for which $ψ$ is not normalizable do not have a direct physical interpretation, but are very useful: suitable superpositions of them give normalizable solutions that can represent a particle. 

However, this page implies that if you perform an energy measurement, the system collapses into the energy eigenstate corresponding to the resultant measured value. http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/superposition/superposition.html

Result of measurement [in QM Statefunction]: The state [$\Psi$] is destroyed. The system falls to one of the eigenstates jn after measurement.

I think that this means that those eigenstates exist but cannot be observed. Is this correct? If they aren't observed, what is their significance?

A:

At the bottom of page 5 and top of page 6 in the linked notes, Zweibach says

For  simplicity  we  discuss  the  case  when  the  spectrum  is  denumerable  so  that,  as  above,  we  can label  the  states  and  energies  with  the  integer  label $n$.  In general  a potential $V(x)$ can  result  in a spectrum that  contains a discrete part  and  a continuous part.  The discrete part  is denumerable but  the  continuous  part  is  not.  The  formulae  we  will  write  below  require  some  modification when  there  spectrum  contains  a  continuous  part (emphasis mine).  The  eigenstates  of  the  continuum  spectrum are  not  normalizable.

You'll notice that the link from GMU says 

The superposition principle states that a statefunction $|\Psi\rangle$ can be expanded as a linear combination of the normalized eigenstates $|\varphi_n\rangle$ of a particular operator that constitute a basis of the space occupied by $|\Psi\rangle$. For the discrete case (emphasis mine):

$$|\Psi\rangle = \sum_{n=0}^\infty b_n|\varphi_n\rangle$$
The GMU site never explicitly treats the continuous case, and as a result, their description of the effects of measurement needs to be modified.  
Ultimately, the resolution of this problem lies in the fact that because measurements have finite resolution, measurement of an observable which has a continuous spectrum (like position, momentum, or, in the case of the free particle on a line, energy) never returns a single real number, but rather some interval $I\subset \mathbb R$.  As a result, the system is projected not into a single (non-normalizable) eigenstate, but rather into a (normalizable) superposition of those eigenstates.  The language is technical, but the Wiki article on the position operator might be helpful; for a more visual picture of how position measurement works, see my answer here.

