A psychologist once told me that people do not begin to dream until they've fallen into a relatively deep sleep. I know her assertion to be false based upon my own experience and upon the reports of others. Some of us begin dreaming right after or perhaps right before falling asleep. Being either barely on one side of consciousness or perhaps in an intermediate state in which dreaming occurs is not quite the same as what is normally called
lucid dreaming, but I'm able to notice some peculiar psychological phenomena as such.
Amongst these are spurious memories. In a dreaming state, I seem to remember events that did not occur in the waking world, though I don't experience those events within the dream. Because the events are not dreamt, but instead there is an apparent memory of these events, it can be harder upon becoming wakeful to discern that the apparent memory were false. But such spurious memories do disintegrate much like memories of dreamt events. In fact, I notice apparent memories often disintegrating within dreams, which disintegration is sufficiently troubling to make me more wakeful.
One disintegration, experienced a few mornings ago, was especially disturbing. I dreamt that I had an old Japanese bank note, and I had (spurious) memories of how I'd acquired the note. But I dreamt that some woman stole the note from me; and, as I dreamt of that theft, my apparent memories of how I'd come to have the note disintegrated, as if themselves stolen.