Gee, we had devices that did that when I went to college. We called them tape recorders.
And yes, professors did permit students to record their lectures.
Nowadays there are many professors and schools which require you to get specific permission to record their lectures, and many of these don't even allow you to record them at all. It's a shame as when I was in college I learned much more when I was able to record my lectures and listen to them as I studied my notes, back then I'd even sleep with a special pillow with speakers inside with my lectures playing back!