Ok thanks. I was aware of the basic theory. I wondered more about the tx filter shape. As we couldn't suddenly cut off at a particular frequency, I assumed, say, 3dB at 3kHz and then so many dB per octave above that. I was wondering if that had been formally defined. Sounds like it was more pragmatic than that then?
Although I have no experience of commercial/broadcast transmitters, I suspect that it may well have been essentially 'pragmatic' (or, at least, 'implicit'). In terms of amateur AM transmitters, I certainly don't recall any explicit steps being taken to limit the 'local' bandwidth - we just relied on the fact that the audio modulating signal would never be more than 2-3 kHz. The only TX filtering we did was on a much wider (frequency) scale, to limit the extent to which harmonics etc. of the modulated RF signal got to the aerial.
Having said that, given that channels with 9 kHz spacing were formally defined/allocated, one imagines that there very probably was also a formal limitation on bandwidth (to ensure that no transmitted signals were more than 9 kHz wide - e.g. if someone were mad enough to try transmitting high frequency data etc. by AM!). However, again, I suspect that may well have usually been achieved essentially 'pragmatically' (on the basis of the maximum likely frequency of the modulating audio), rather than by explicit filtering.
If it were done, I presume that it would be the modulating signal (audio) that would have had a low-pass filter, since it would be infinitely easier to create a filter with a rapid fall-off after say, 3 or 4 Hz than it would to achieve the same with the modulated RF signal. Even 'back then', it was quite feasible to create (even with discrete components!) an audio frequency low-pass filter that had very rapid fall-off beyond a particular frequency.
AM is, of course, an inherently inefficient means of communication, both in terms of power usage and bandwidth required - much of the power is in the carrier, which carries no information, and one of the sidebands is redundant (hence making the bandwidth twice as wide as it needs to be), since it is merely a mirror image of the other sideband. That is why 'SSB' is preferable (strictly 'SSBSC' - "Single Sideband with Suppressed Carrier") - occupying half the bandwidth and wasting no power in the carrier. Most of the ways in which SSB is generated would also, as a 'side effect', limit the bandwidth of the one 'surviving' sideband (i.e. 'the signal').
Kind Regards, John