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?